There isn't much to design. One border case is the λ/4 rod over an infinite ground plane, it has a nominal impedance of 40 ohm, the other a λ/2 dipole with 73 ohm impedance. In an more exact analysis, there is a certain dependence of impedance and optimal length on rod diameter. In most cases, one uses a thick wire or thin tube, that can be still assumed as ideal wire in analysis.
A real mobile instrument with monopole antenna and sufficient ground plane size will range somewhere between both sketched border cases and most likely show an additional reactive impedance part depending on the ground plane size. Ideally, empirical impedance matching would be applied. As the antenna impedance is at least near to 50 ohms transceiver impedance, just connecting the antenna without individual matching network is often the obvious solution.