The evolution of nonlinear resonant oscillations in closed tubes: Solution of the standard mapping

The concern of this paper is the evolution of small-amplitude resonant oscillations of an inviscid gas in a closed tube. The evolution of the oscillations of the gas generated at and near the fundamental frequency and half the fundamental frequency, where experiment shows that shocks are a feature of the final periodic motion, is examined. The basis for the analysis of this nonlinear initial value, boundary value problem on a semi-infinite strip is the Dissipative Standard Map. Since the purpose is to elucidate how the initial rest state of the gas evolves to the final periodic state, the focus of the analysis is on how a prescribed initial curve is mapped, under the Dissipative Standard Map, onto an invariant curve. The method used is to approximate the Dissipative Standard Map, in the small rate limit, by the partial differential equation appropriate to the resonance in question.