Nonlinear forced oscillations in a closed tube: continuous solutions of a functional equation

Abstract The continuous small amplitude disturbances generated by the oscillations of a piston in a gas contained in a closed-ended tube are discussed. The coupled characteristic equations are integrated exactly for a model equation of state which approximates any stress–strain law with an error at O([strain]3). When the small amplitude limit is taken and the relative importance of amplitude dispersion and nonlinear interaction is assessed, the disturbances can be determined from solutions to a nonlinear functional equation. The motions are characterized by a similarity parameter, A, and a frequency parameter, ∆. A simple algebraic scheme for constructing continuous periodic solutions of the functional equation is given, and the A–∆ plane is divided by a transition curve. The various resonances are then evident.