Nonlinear asymptotic impedance model for a Helmholtz resonator of finite depth

A systematic derivation of the solution of a nonlinear system of equations for the finite-length Helmholtz resonator is presented, modeling the configuration of an organ-pipe type cavity (i.e., of finite length) connected via an acoustically small neck to the external excitation field. The model of the flow through the neck includes linear viscous friction and nonlinear dissipation due to vortex shedding. By assuming a weakly nonlinear amplitude regime (i.e., essentially nonlinear near resonance but effectively linear away from resonance), it is possible to set up a solution, asymptotic for small excitation amplitudes, which enables analytically obtaining an expression for the impedance that includes nonlinear effects for frequencies close to the fundamental resonance frequency. (The higher harmonics will not be considered). This paper extends and refines the previous analysis that considered an acoustically compact cavity. Apart from a confirmation of the previously published impedance results in the small cavity limit, the new results establish a significant improvement in the comparison with experimental data and the asymptotic matching of the linear and nonlinear impedance regimes. This is in part due to the obviously refined model of the geometry but also due to the equivalent but numerically better alternative form of the asymptotic expressions that, by chance, emerged in the new analysis. Although both versions are asymptotically equivalent to the order considered, the new form happens to behave better for larger values of the excitation amplitude.