The transient excitation of a straight thin wire segment parallel to a plane interface between two homogeneous dielectric half spaces is analyzed by the continuous-time, discretized-space approach. The analysis is carried out in three steps. First, the subproblem of a single wire embedded in a homogeneous dielectric medium, excited by a voltage source and an incident electric field, is studied with the aid of a time domain integral equation. This equation is discretized in space and transformed to the frequency domain. This procedure results in a system of linear equations of a fixed dimension which is solved by marching on in frequency. Second, the subproblem of a horizontal, pulsed dipole over the interface between two homogeneous dielectric half spaces is considered. The reflected field in the upper medium is computed by spectral techniques. With the aid of a time domain Weyl representation, a fixed, composite Gaussian quadrature rule is derived for the semi-infinite integral involved. The angular integral is evaluated in closed form. Third, for the complete problem, the reflected field in the upper medium is expressed as a superposition of contributions from dipole sources on the wire axis, and subsequently treated as a secondary incident field in the integral equation for the current on the wire. This integral equation is then solved by combining the techniques developed for the two subproblems. Representative numerical results are presented and discussed.