Time-dependent plane-wave spectrum representations for radiation from volume source distributions

Abstract

A new time-domain spectral theory for radiation from a time-dependent source distribution, is presented. The full spectral representation is based on a Radon transform of the source distribution in the four-dimensional space-time domain and consists of time-dependent plane waves that propagate in all space directions and with all (spectral) propagation speeds vκ. This operation, termed the slant stack transform, involves projection of the time-dependent source distribution along planes normal to the spectral propagation direction and stacking them with a progressive delay corresponding to the spectral propagation speed vκ along this direction. Outside the source domain, this three-fold representation may be contracted into a two-fold representation consisting of time-dependent plane waves that satisfy the spectral constraint vκ=c with c being the medium velocity. In the two-fold representation, however, the complete spectral representation involves both propagating time-dependent plane waves and evanescent time-dependent plane waves. We explore the separate role of these spectral constituents in establishing the causal field, and determine the space-time regions where the field is described only by the propagating spectrum. The spectral theory is presented here for scalar wave fields, but it may readily be extended to vector electromagnetic fields.