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

Abstract

The zero-ordetre rm of the time domains catterede lectricf ieldo f an electromagnetic plane wave normally incidentu pon the surfaceo f two quarter spacesis determined. The generals olutioni s a developmentf rom a previouse xact and completes olutioni n the frequency domain. The zero-order term of the scattered electric field has been computed in the upper medium( z < 0). The incidentw avei n the frequencyd omaina ssumesth e samef unctionf or threec ases:( 1) The conductivitvya nisheesv erywhere(2; ) only the conductivityo f the upperm ediumi s zero;a nd (3) the three mediaa re conductorsC. ase 1 helpst o understandc ases2 and 3. Case 2 is applicablet o geophysicael xploration. For cases 1 and 2 a causal time function decaying exponentially with time at every point above the fault (z < 0) describesth e waveformo f the incidentp lanew ave. The zero-ordert erm of the scattered field has been computed above the fault. At z = 0 it reduces to a closed expressionf or case 1 and to a singlei ntegral for the other two cases.I n the three casesi t contains an integral of a Hankel function for z • 0. The computation of the high-frequency part of the inverseF ourier transformf or z -• 0 employsa symptotice xpressionsfo r the Hankel function usinga nalytical techniqueso f the geometricalt heory of diffractionf or 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 w plane. The wave front of the scattered field is well defined in shape, phase, and amplitude. Its amplitude is discontinuouast ß - 0, and variess moothlyb ut presentsa sharpj ump for I• I<<1 z I. For I z I: O(z), there is a numericanl oiset hat oscillatesa t 100M Hz.