In this paper, a time-domain variant of the generalized telegrapher's equations for transient electromagnetic field coupling to a finite-length wire above a lossy half-space is derived. The approach is fully based on the thin-wire antenna theory. The lossy ground effects are taken into account by means of the reflection coefficient approximation. The obtained equations are handled numerically via the Galerkin-Bubnov indirect boundary element method. Computational examples are presented for the case of a single-wire line excited by an electromagnetic pulse excitation source. The obtained results for the induced current along the line are compared with those obtained using 1) the method of moments solution of the electric field integral equation implemented in the numerical electromagnetics code (NEC-4), and 2) the transmission line (TL) theory. It is shown that the results obtained by the proposed method are in excellent agreement with those of NEC-4. It is also shown that the TL approximation yields in general results which are in reasonably good agreement with the full-wave results, especially for the early time response and even beyond the limits of the accuracy of the TL theory. The TL theory can, however, give accurate results only for times before the arrival of the first reflection from the TL terminations and it fails to reproduce accurately the dispersion effects occurring after the first reflection.