Spectral alternatives for the synthesis of short-pulse wavefields in waveguides

For the synthesis of short-pulse 2D wavefields in simple waveguiding configurations, various alternative spectral field representations can be constructed. In the Spectral Theory of Transients (STT), a wavefield is expanded in terms of non-dispersive time-domain plane wave constituents. The STT furnishes an appropriate setting for the description of short-pulse fields within the Fresnel zone in a waveguide. In the far zone a ray-field expansion is not very effective, since the constituents cease to be well resolved in space and time. Therefore, a global field expansion in terms of a frequency spectrum of time-harmonic modes is more appropriate, but does not yield sufficient information about the time behavior of the field. Fortunately, the scope of the STT can be extended through the construction of an angular spectrum of global dispersive time-domain spectral-mode (TDSM) constituents, which exploits the advantages of both synthesis options. The field representation in terms of a spectral integral of a series of TDSM constituents proves to be equivalent to the representation in which the field is assembled with contributions from a periodic infinite array of image sources. The use of analytic signals allows for the complication of the space-time source coordinates in the resulting expressions for the field, thus yielding the 2D complex-source pulsed-beam (CSPB) solutions, associated with a class of directive sources. The asymptotic evaluation of the remaining spectral integral leads to an expression for the modal field in terms of instantaneous frequencies and instantaneous angles of propagation, through which a cogent description of the space-time-domain wave phenomena is obtained.