Input impedance change of a half-wave vertical antenna over a dissipative earth

The problem investigated here is that of calculating the input impedance of a thin vertical center-fed half-wave dipole over a dissipative ground. A variational expression for the input impedance is found which is valid for heights above the ground as small as (in many cases the limit is smaller than this) and for refractive indices such that . Using a zeroth-order trial function, a closed-form expression which represents the change in the input impedance of the antenna between the case of a perfectly conducting ground and the case of a dissipative ground is found.