Time-domain asymptotics. I General theory for double integrals

Abstract

A technique is described for obtaining asymptotic formulae for the time-domain waveforms associated with double integrals of rapidly-varying isolated pulses. The method is an extension of the theory of the time-domain asymptotics of single integrals given by C. J. Chapman ( Proc. R. Soc. Lond. A 437, 25–40 (1992)), and represents a generalization of the method of stationary phase to integrals with non-sinusoidal integrands. Stationary points of the phase function in the domain of integration each give rise to a particular waveform shape in the time-history of the solution; the waveforms associated with interior extrema and saddle points are proportional to the original source function and the Hilbert transform of the source function respectively. Boundary stationary points also give rise to distinctive waveforms, proportional to fractional integral transforms of the source function. The transitional case of the coalescence of two interior stationary points is considered in some detail; an asymptotic formula describing the coalescence is found, and the limiting behaviour of this formula after coalescence is calculated, i. e. the residual waveform after the annihilation of the stationary points. Asymptotic and numerical results are compared for an example integral, and good agreement is found even for moderate values of the asymptotic parameter.