Interaction between a transient plane wave and a conductive half-space.

A method which allows for the analytical evaluation of the interaction between a transient plane wave and a conductive half-space is presented. We assume that an electromagnetic plane wave is obliquely incident on a conducting half-space, which is modeled by a frequency independent permittivity and conductivity. The general case of the electromagnetic plane wave is divided into two polarizations: transverse electric (TE) and transverse magnetic (TM). The time-domain expressions for the reflected and transmitted waves are first represented as inverse Laplace transforms. The transient fields are then shown to consist of two canonical integrals, f(β) and e(β)The canonical integrals, in turn, are solved analytically, thereby yielding closed-form solutions involving incomplete Lipschitz-Hankel integrals (ILHIs). The ILHIs are computed numerically using efficient convergent and asymptotic series expansions, thus enabling the efficient computation of the transient fields. The exact, closed-form expressions are verified by comparing with previously published results and with results obtained using standard numerical integration and fast Fourier transform algorithms. An asymptotic series representation for the ILHIs is then employed to obtain a relatively simple late-time approximation for the transient fields. This approximate late-time expression is shown to accurately model the fields over a large portion of their time history. The closed-form transient field expressions are used to investigate the effects of neglecting displacement currents when studying transient wave propagation in the conductive half-space. The diffusion fields are found to yield accurate results at late times