Nonlinear Oscillations in a Resonant Gas Column: An Initial-Boundary-Value Study

Acoustic resonance in a gas column driven by a vibrating piston is studied in terms of an initial-boundary-value formulation. If the boundary opposite to the piston face permits acoustic energy leakage, the linear solution prevails and describes the evolution to periodic, quasi-steady oscillations at the piston frequency. Resonant amplification of the wavefield may occur only if the opposite boundary is either closed (a rigid wall) or ideally open (an isobaric exit). A multiple-timescale perturbation analysis, based on Fourier eigenfunction representations, is used to find weakly nonlinear solutions that provide unique physical insights in terms of acoustic modal interactions and transient waveform evolution to limit cycles. It is shown that quadratic modal interactions in a closed system lead to persistent shock appearance and relatively small limiting amplitudes, whereas cubic mode-coupling dominates in an open system and stronger shocks are created intermittently. Finite-difference solutions support these conclusions. The initial-value approach employed enables the prediction of shock waves in parameter regimes where no shock has been found by earlier investigators using a quasi-steady formulation.