Time-domain asymptotics and the method of stationary phase

Abstract

A technique is described for obtaining the asymptotic behaviour at large λ of integrals having the form I ( λ ) = ∫ g(x) p{λf(x)} dx , where p is an arbitrary periodic function with mean zero. It is based on the fact that the method of stationary phase may be applied directly to p { λf(x) }, without decomposition of p into Fourier components: thus a simple minimum of f at x = x 0 gives a term containing a fractional integral of order one-half, proportional to p̂ + ( y ) = { ∞ y { p(x)/ (x — y) 1/2 } d x evaluated at y = λf(x 0 ) ; a simple maximum gives a similar term. In many physical problem, f depends linearly on a parameter, say t , in such a way that I is periodic in t and the quantity of interest is q(t) = d l /d t . The theory of how the shape of q is determined by that of p when λ is large but fixed is here called waveform asymptotics and its main features investigated using ‘barber’s pole’ integrals. For example, the singularity in q produced by a discontinuity in p is found explicitly as an inverse square-root multiplied by a coefficient, so need not be inferred from the tail of a Fourier series. More generally, the effect on q of any rapid change in p may be obtained by the present method of stationary phase in the time domain, without resolution into components; since Gibbs’ phenomenon is thereby avoided the method is suited to highly non-sinusoidal wave problems. An asymptotic representation of q by zeta functions is possible. Four extensions of the basic theory are analysed in detail: coalescence of a maximum and minimum of f ; contributions from the end-points of the range of integration; collision of a maximum or minimum with an end-point ; and the behaviour of integrals with no stationary points or end-points. The first and third of these lead to time-domain Airy functions and Fresnel integrals, respectively, with singularity structures dual to the smooth patterns found in diffraction catastrophes; the second recovers the original waveform; and the fourth gives exponential asymptotics. The theory is illustrated throughout by analysis and computation for functions p describing a square wave and an intermittent N -wave, and by diagrams of the resulting waveforms.