Time-domain asymptotics and the method of stationary phase

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A technique is described for obtaining the asymptotic behaviour at largeλof integrals having the formI(λ) = ∫g(x) p{λf(x)} dx, wherepis an arbitrary periodic function with mean zero. It is based on the fact that the method of stationary phase may be applied directly top{λf(x)}, without decomposition ofpinto Fourier components: thus a simple minimum offatx = x0gives a term containing a fractional integral of order one-half, proportional top̂+(y) = {∞y{p(x)/ (x — y)1/2} dxevaluated aty=λf(x0); a simple maximum gives a similar term. In many physical problem,fdepends linearly on a parameter, sayt, in such a way thatIis periodic intand the quantity of interest isq(t)= dl/dt. The theory of how the shape ofqis determined by that ofpwhenλis large but fixed is here called waveform asymptotics and its main features investigated using ‘barber’s pole’ integrals. For example, the singularity inqproduced by a discontinuity inpis 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 onqof any rapid change inpmay 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 ofqby zeta functions is possible. Four extensions of the basic theory are analysed in detail: coalescence of a maximum and minimum off; 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 functionspdescribing a square wave and an intermittentN-wave, and by diagrams of the resulting waveforms.