Time-Domain Modes — Asymptotic Expansion and Error Estimates

Abstract

The spectral theory of transients (STT) comprises a ray-field expansion of a wavefield in terms of non-dispersive time-domain plane-wave constituents. In a waveguide, the STT furnishes an appropriate setting for the description of short-pulse fields within the Fresnel zone. Beyond the Fresnel zone the constituents in a ray-field expansion cease to be well resolved in space and time, rendering the resulting expansion less effective. The scope of the STT can been extended through the construction of an angular or transverse spectrum of a series of global time-domain spectral-mode (TDSM) constituents, in which the dispersive temporal behaviour of short-pulse fields is explicit.

The remaining integral of TDSM constituents over the transverse spectrum is amenable to asymptotic evaluation. The resulting asymptotic field description is not only numerically expedient, it also provides a cogent description of the underlying physics involving instantaneous frequencies and angles of propagation. The moment at which the signal changes from chirping up to chirping down in time is close to the arrival time, while the duration of the coda of the signal is typically quite long. Hence, the response to a finite pulse predominantly chirps down in time.

The leading two terms in the asymptotic expansion may be used to determine an asymptotic measure for the truncation error. This error decreases monotonically for increasing mode number and increasing time, such that for any set error bound an instant in time may be determined beyond which all time-domain modal field constituents can be evaluated asymptotically to within that error bound.

For impulsive sources, the field becomes singular at every arrival. Upon evaluating the time-domain modes asymptotically, the corresponding singular behaviour in the modal series may be regularised. The resulting expression is amenable to rational approximation, yielding an expedient and accurate approximation to the time-domain Green’s function in a waveguide.