Zero-order time domain scattering of electromagnetic plane waves by two quarter spaces

The zero-order term of the time domain scattered electric field of an electromagnetic plane wave normally incident upon the surface of two quarter spaces is determined. The general solution is a development from a previous exact and complete solution in the frequency domain. The zero-order term of the scattered electric field has been computed in the upper medium (z < 0). The incident wave in the frequency domain assumes the same function for three cases: (1) The conductivity vanishes everywhere; (2) only the conductivity of the upper medium is zero; and (3) the three media are conductors. Case 1 helps to understand cases 2 and 3. Case 2 is applicable to geophysical exploration. For cases 1 and 2 a causal time function decaying exponentially with time at every point above the fault (z < 0) describes the waveform of the incident plane wave. The sere-order term of the scattered field has been computed above the fault. At z = 0 it reduces to a closed expression for case 1 and to a single integral for the other two cases. In the three cases it contains an integral of a Hankel function for a: not equal 0. The computation of the high-frequency part of the inverse Fourier transform for a: not equal 0 employs asymptotic expressions for the Hankel function using analytical techniques of the geometrical theory of diffraction for cases 1 and 2. For case 3 the inverse Fourier transform may have two possible contributions: either from the residue at a single pole or from the integral along a branch cut in the omega plane. The wave front of the scattered field is well defined in shape, phase, and amplitude. Its amplitude is discontinuous at a:= 0, and varies smoothly but presents a sharp jump for \ x \<<\ z \. For \x\= O(z), there is a numerical noise that oscillates at 100 MHz.