The wavelength of a note is the speed of sound divided by the pitch (the frequency of vibration of the air). The wavelength gets shorter as the note gets higher, which is very evident in an instrument like the Pan Pipes, where shorter tubes produce higher notes. An instrument also has a timbre, which describes the level of harmonics produced when playing a note. Harmonics are multiples of the pitch. (Tuning a string, by changing the tension, actually changes the speed of sound in the string, hence the pitch and wavelength for the same string length )
Antennas use the same language, as they are also dealing with vibrations. In the antenna case, the vibration is an electromagnetic wave, rather than a sound wave in air molecules. The speed of electromagnetic radio signals is the speed of light, rather than the speed of sound. Light being an electromagnetic (radio) wave that we can see (Our eyes don't have antenna that can see other radio waves).
For antennas, We don't refer to notes or pitch, but use the term "frequency". Frequency is measured in Hertz, though the older term, cycles per second, was more descriptive. The wavelength of a radio signal is the speed of light divided by the frequency, so, like sound, the wavelength gets shorter as the frequency increases. Hence our antennas get smaller as the frequency increases. Radio signals also produce harmonics, which are dependent on the way we add the digital messages to the WiFi carrier (or center) frequency.
A wire antenna is like a vibrating string or tuning fork. The antenna wire is cut to the wavelength (more often a sub-harmonic of the wavelength) and will resonate when there is a radio signal arriving, and this will produce an electrical signal in the wire that the radio can use. The reverse happens when we transmit.
A waveguide antenna is more like a Pan Pipe equivalent. The internal dimensions of the waveguide antenna are made so the waveguide is resonant at the frequency of interest. The slots we cut in the face of the waveguide are also cut to resonate at the frequency of interest. A transition is needed to get from vibration in the waveguide, to vibration in the coax going to radio. This is usually a small wire antenna, called a probe. We have also had good results from having a cone shaped probe.
A wire antenna resonates with the electrical part of the electromagnetic wave, and the slots in the waveguide resonate with the magnetic part of the electromagnetic wave. The electric and magnetic parts of the electomagnetic wave are at right angles to each other, so a horizontal wire is equivalent to a vertical slot. Antennas will work with frequencies close to the carrier signals frequency. How efficently, depends on the antenna design. This is refered to as the antennas bandwidth.
Our original antennas are all horizontally polarized (the electric component being horizontal, and the magnetic compenent vertical). The newer commercial basestations have dual polarization, also having both vertically polarized and horizontally polarized patch antennas. This is an advantage, as the path loss (between the two radios) can be different for horizontal and vertically polarized signals.
It should also be noted that the speed of light in wire is lower than the speed of light in air. The material around the wire (even copper oxide) effects this velocity of a signal in the wire too, pushing down the resonant frequency. This is represented by the velocity factor, which is the percentage of the speed of light in the antenna or cable compared to the speed of light in air. This is called the velocity factor. Cables also have there own velocity factor, which depends on how they are made.
The velocity factor of copper wire is about 0.95 times the speed in air, but it does vary with different wire. This makes a copper antenna about 5% shorter than Free Space wavelength of the signal. The diameter of the wire is also a factor. For radio waves, the effect is due to induced movement of electrons in the material. This is the same reason light appears to be slower in glass, than in air, yet speeds up again when leaving the glass. The induced signal in the glass gets added to the arriving signal, and the resulting wave has a slower speed than light in air.
Impedance is is a term you will see a lot. It is the AC equivalent of DC resistance, and both are measured in Ohms. For radio waves, this is usually 50 Ohms for the radio, but not always, and for air is approximately 377 Ohmns. Physically, impendance manifests as the ratio of the magnitude of the electic and magnetic fields.
A mismatch between between the impedance of the Antenna (including in the cables) and the impedance that the transmitter expects will result in a reflection of some of the transmitted energy back to the transmitter. At visual radio wavelengths (light), a mistmatch from air to glass can be seen as the reflection of light off of a window. About 4% of the light is reflected, due to the difference between the impedance of air and glass. At high power levels, this reflected energy can cause damage to the transmitter, so testing should be done at low transmit levels. Most wifi radios are low enough power, to not be in danger, but a mismatched impedance still reduces the efficiency for both transmitting and receiving.
Testing the efficiency of receiving, by using a remote WiFi signal, will give you a good indication of the transmitting efficiency of the antenna. Antennas are mostly symmetric for receive and transmit. This is called repricosity. Getting a perfect match between the antenna and radio is unusual. There are antenna testers (SWR meters (Standing Wave Ratio)), that can be used to tell how much of the transmitted signal is being reflected. There are also testers that measure the impedance and bandwidth (though many testers do not work at WiFi frequencies). An SWR below 1.5 is the goal, with Ratio of 1 being ideal. Some antenna need a matching network, but none of our designs use them.
Cable loss is a significant factor too. Cable loss at 2.4GHz is significant, and much worse at 5GHz, so keep antenna cables very short. You can find cable loss from the manufacturers web sites, and there are some online calculators. LMR100 cable is thin, but you should keep the length very short, as it has high loss per meter. LMR200 has about half the loss of LMR100, and LMR240 slightly less. LMR400 has about half the loss of LMR200, but is quite thick. We did use up to 2m runs of LMR400, but eventually connected to our waveguides with right angle N-connectors, directly to our WiFi routers, or used very short LMR100 (100mm) jumpers to the WiFi Routers. Connector also have a small loss, which is usually documented by the manufacture.
Cable loss will also affect the SWR reading of a test, as the reflected wave will experience loss coming back down the cable. The SWR will then look to be lower than it really is. Test the antenna, with a short calibrated cable, and the separately test the feed cable, using a dummy 50 Ohm resister, instead of the antenna.
Keep the antenna away from other metal objects. At least a wavelength is advisable, which for 2.4GHz is only 123mm, so not very far. You also want to minimise reflections from behind, or from objects in the signal path. I did have a location with a large semicircle of rock behind the house, and despite getting a good signal, the wifi wouldn't work due to reflections causing interference with the primary signal. Moving the antenna only a small distance solved the problem. We have also found pine trees near the signal path, to act like tinsel reflecting the radio wave (their Pine needles being about the length of a 2.4GHz antenna and absorb and reradiate the signal).