The evolution of resonance: a multiscale approach to the effect of nonlinearity, frequency dispersion and geometry

Abstract

Nonlinear resonant oscillations in continuous media contain two natural time scales: the travel time in the medium and the ‘slow’ time defined by the small nonlinearity. A multiscale approach is used to describe the evolution to a periodic state. We focus on three basic experiments that define nonlinear resonant oscillations in continuous media: a gas in both a straight tube and a tube of variable cross-section, and shallow water in a tank. The outcomes of these experiments are described and the mathematical techniques that show the evolution to the final periodic states are given in some detail.