Nonlinear asymptotic impedance model for a Helmholtz resonator liner

The usual nonlinear corrections for a Helmholtz resonator type impedance do not seem to be based on a systematic asymptotic solution of the pertaining equations. We aim to present a systematic derivation of a solution of the nonlinear Helmholtz resonator equation, in order to obtain analytically expressions for impedances close to resonance, while including nonlinear effects. The amplitude regime considered is such that when we stay away from the resonance condition, the nonlinear terms are relatively small and the solution obtained is of the linear equation (formed after neglecting the nonlinear terms). Close to the resonance frequency, the nonlinear terms can no longer be neglected and algebraic equations are obtained that describe the corresponding nonlinear impedance. Sample results are presented including a few comparisons with measurements available in the literature. The validity of the model is understood in the near resonance and non-resonance regimes.