An asymptotic model for non-linear Helmholtz resonator of finite depth

The nonlinear corrections for a Helmholtz resonator type impedance based on systematic asymptotic solution of the pertaining equations has been formulated in [1]. The length of the cavity considered was much smaller compared to the typical acoustic wavelength of interest so that the pressure inside could be assumed to be uniform. This way, the cavity worked as a spring to the external excitation and the velocity obtained was given by the time derivative of the uniform pressure. In the current work, this model is refined to take into account the development of the standing waves inside the cavity so that the model captures more physics of the damping phenomenon inside the cavity. We aim to present a systematic derivation (from first principles) of solution of the nonlinear Helmholtz resonator equation with the neck connected to an organ pipe type long cavity to have the acoustic waves developed inside, in order to obtain analytically, an expression for the impedance close to resonance, while including nonlinear effects. The amplitude regime is considered 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. Apart from a confirmation of the previously published impedance results [1], a better comparison with experimental data and the asymptotic matching of the linear and nonlinear impedance regimes is established.