A so-called random wire antenna is an end fed antenna. One end goes straight into the rig, often with no feedline, and the other end in the air attached to something as high as you can find, as described on the ARRL random wire page. The Wikipedia Electrical Length page has this very nice animation of a center fed dipole.
Notice how voltage (red) is always low at the center, a great attraction of center fed dipoles, but when the received signal is strongest the voltage at the ends is at its highest. Additionally, matching a large capacitive or inductive impedance can generate high voltage in the matching network. Unless the matching network is made of high voltage components impedance cannot be matched. When hardware can't be changed, the end fed solution is to move away from the high voltage. In other words, choose (cut wire) to receive a weaker signal so that the antenna tuner can successfully match. It is important to use a counterpoise with an end fed. A standard recommendation for the 40m band (see QST, March 1936, p. 32, "An Unorthodox Antenna") is an 84' long end fed and a 17' long counterpoise (6.5' for 20m). While these lengths have been shown to work well on many bands, which is helpful if you're in a hurry to get on the air, read up on the topic and experiment.
Other solutions are a more capable tuner or a better matched antenna. See AA5TB's excellent page for EFHW info. Short of that, cut the wire so that it is not a halfwave multiple of any frequency you want to use and accept a weaker signal so that you can impedance match.
For a given ham band its lower and upper band edge frequencies are converted by the famous 468/f to dipole lengths. QST, Oct 1926, "The Length of the Hertz Antenna" shows in line 2 of the table that a similarly installed random wire should use 423/f. W3EDP's results in the above referenced 1936 article, however, fall in line with 468/f and is used here. Change the program to use 423 if desired.
While these are the best lengths for center fed dipoles, they are typically the hardest to match lengths for end feds and the lengths should be avoided.
I wrote a program to do the light number crunching considering all band edges of interest. Again, optimal lengths for strongest signal are also the hardest, or worst, lengths for end-fed impedance matching. For multiple bands, harmonics are also considered. In addition, the wire should be at least 1/4 wavelength of the lowest desired frequency. The program simply calculates length from the venerable L=468/f for band edges and their multiples, for all bands specified.
The graph ranges from 1/4 to 1/2 wavelength by default since you must use at 1/4 of longest wavelength and there is no need for more than 1/2 wavelength of wire. An option is shown to graph a full wavelength, however. You can move your mouse over the graph and the length is displayed as 'x=###' in the upper right.
Download endfed.py and run as shown below. (The anaconda python distribution is an easy way to put python and many libraries on your machine.)
Modify bands in endfed.py for your country or favorite modes. Use it by putting space separated band names and a graph will be created. Sample runs are shown:
endfed.py 40 20 15 10
(half wavelength, feet).
endfed.py -f 40 20 15 10
(full wavelength, feet).
Other possible ways to run:
| endfed.py -m 40 20 15 10 | (metric). |
| endfed.py -c -m 40 20 15 10 | (CW sub-bands, metric). |