An Asymptotic Theory for Oscillatory Non-linear Signals

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Abstract
A general method is developed for finding approximate “asymptotic” solutions to a large class of non-linear partial differential equations. Solutions are presented for signals having one predominant mode, with a characteristic length scale much shorter than all other lengths characteristic of the signal geometry or of the medium being traversed. Non-linear distortion of a signal of arbitrary initial waveform is described, in addition to the relaxation damping and modulation as it penetrates disturbed or inhomogeneous regions. Equations also are derived showing how the signal drives its “background”, forcing this slowly to drift. The asymptotic procedure extends the basic concept of “relatively undistorted waves” (Courant & Hilbert, 1965; Varley & Cumberbatch, 1970) using an approach applied initially to the modulation of non-linear pulses (Parker & Varley, 1968). For disturbances of restricted amplitude this description gives a “non-linear acoustics”, allowing profile distortion and shock formation.

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