A finite-rate theory of resonance in a closed tube: discontinuous solutions of a functional equation

Abstract

The only solutions to date which describe nonlinear resonant acoustic oscillations are those for which the distortion of the travelling waves is negligible. Many experiments do not comply with this small-rate restriction. A finite-rate theory of resonance for an inviscid gas, in which the intrinsic nonlinearity of the waves is taken into account, necessitates the construction of periodic solutions of a nonlinear functional equation. This is achieved by introducing the notion of a critical point of the functional equation, which corresponds physically to a resonating wavelet. In a finite-rate theory a wave may break in a single cycle in the tube, and thus there may be more than one shock present even at fundamental resonance. Discontinuous solutions of the functional equation are constructed which satisfy the weak shock conditions.