While experimenting with different node setups, I had the need for a couple of antennas. The information I found on the internet for 868 MHz antennas did not prove accurate, so for completeness I include the calculations for antenna lengths for LoRa applications in the 433 and 868 MHz band.
An antenna is a conductor in the form of a spline, electrically connected to the communications module through a transmission line. The antenna diameter hardly has any importance, as long as the antenna stays in the spline form. The most effective antenna has the same length as the length of the wave it is used for. For practical purposes, half or a quarter of that length will suffice. Most LoRa antennas are a 1/4 wavelength.
The wavelength of a frequency is calculated as v / f, where v is the speed of the transmission and f is the (average) transmission frequency. In air, v is equal to c, the speed of light, which is 299.792.458 m/s. The wavelength for the 868 MHz band is thus 299.792.458 / 868.000.000 = 34,54 cm. Half of this is 17,27 cm and a quarter is 8,63 cm.
For the 433 MHz band the wavelength is 299.792.458 / 433.000.000 = 69,24 cm. Half of this is 34,62 cm and a quarter is 17,31 cm.
A piece of wire of 8,6 cm therefore will do for a LoRa application in the 868 MHz band. The exact length is a major factor in the quality of an antenna. Unless the antenna is directly soldered to the LoRa module, use 50 ohm cable and certified connectors for any transmission line.