The Basics of Calculating Wavelength

In order to gain an understanding of how antennas work, I think one needs to have some knowledge about capacitors and inductors. If an antenna is short, (by this I mean shorter than required for resonance) it acts a bit like a capacitor. We say the antenna is capacitive. If the antenna is long or longer than it would be at resonance it acts like a coil or inductor. We usually say that an antenna is capacitive in the first case and inductive in the second case.
It naturally follows that it is important to know if the antenna is resonant or how long would it need to be to be resonant before we can decide if it may be short or long. (This is where I would like to have a blackboard) When dealing with radio frequencies the distance the electrical signal (in a conductor) or radio wave (same signal after it leaves the antenna) travels in one cycle is the key to everything.
If we are using a frequency of 10 MHz, that means that the radio frequency signal goes through ten million complete cycles in one second. The signal will travel 30 meters in the time of one cycle. The wavelength is 30 meters. Wavelength is simply how far the signal will travel in the time it takes to complete one cycle.
For the time being we will only consider wire antennas. Other larger conductors will be discussed later after the fundamentals are established.
The shortest length of wire that is considered resonant is one half wavelength. The free space wavelength is easy to establish. The speed of the wave divided by the frequency will give the wavelength. There are several well known formulas you can use depending on what units you want to use. If you want to get your answer in meters you divide 300,000,000 by the frequency in Hertz. Another variation on that formula is to divide 300 by the frequency in MHz. Either way the answer is in meters. To get your answer in feet (I measure all my antennas in feet) you divide 982,000 by the frequency in Hertz . This is again for one wavelength. For a half wavelength you simply divide that answer by two. In the half wavelength case the formula simplifies to: Half wavelength = 491,000/ frequency in Hertz.
Also by dividing 491 by the frequency in MHz gives you a half wavelength in feet. I suggest that this is a formula to memorize. There is also another similar formula in all the antenna books using 468. It is simply a special case based of the 491 number. We will use 491 in calculations and then, depending on what we are doing, may multiply it by something like .95 or 95% . This is because the speed of a radio wave or signal is slower in a wire or coax transmission line than in free space and we need to adjust our formula accordingly. When working with transmission lines it is important to use the correct velocity factor.
We have been using 186,000 miles per second and 300,000,000 meters per second as the speed of radio waves (speed of light). These are good numbers to memorize if you do not already know them. By the way the 982,000 number comes from multiplying 186,000 Miles/second by 5280 feet/mile to get 982,000 feet/second.