## 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.

## Footnotes

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- Received June 27, 1991.
- Accepted December 3, 1991.

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