## Abstract

A technique is described for obtaining the asymptotic behaviour at large $\lambda $ of integrals having the form I($\lambda $) = $\int $ g(x)p{$\lambda $f(x)} dx, where p is an arbitrary periodic function with mean zero. It is based on the fact that the method of stationary phase may be applied directly to p{$\lambda $f(x)}, without decomposition of p into Fourier components: thus a simple minimum of f at x = x$_{0}$ gives a term containing a fractional integral of order one-half, proportional to $\hat{p}_{+}$ (y) = $\int_{y}^{\infty}${p(x)/(x-y)$^{\frac{1}{2}}$} dx evaluated at y = $\lambda $f(x$_{0}$); a simple maximum gives a similar term. In many physical problems, 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) = dI/dt. The theory of how the shape of q is determined by that of p when $\lambda $ 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 endpoints of the range of integration; collision of a maximum or minimum with an endpoint; 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.

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