Scattering to Higher Harmonics for Quarter-Wave and Helmholtz Resonators

The nonlinear response of acoustic resonators is investigated over a broad range of frequencies and amplitudes. Helmholtz resonators with a symmetric neck and an asymmetric neck, respectively, as well as quarter-wave resonators are considered. Describing functions for impedance and the reflection coefficient of a Helmholtz resonator at various sound pressure levels are determined from compressible flow simulation and validated against experimental data. The particular focus of the present study is the nonlinear scattering to higher harmonics. For the Helmholtz resonator with a symmetric neck, a distinct pattern in the amplitudes of the higher harmonics is observed, where the odd harmonics dominate the response, whereas the even harmonics are almost negligible. Such an "oddharmonics-only "pattern, which was observed previously in an experiment at the orifices, is explained by a quasi-steady analysis based on the Bernoulli equation, assuming a symmetric flow pattern at the neck. For the Helmholtz resonator with an asymmetric neck, it is observed in computational fluid dynamics simulations that even harmonics contribute noticeably to the resonator response, such that the odd-harmonics-only pattern is less pronounced. For the markedly asymmetric geometry of the quarter-wave resonator, the second harmonic is dominant and the odd-harmonics-only pattern vanishes completely. The quasi-steady analysis is extended successfully to also describe nonlinear scattering to higher harmonics for asymmetric configurations and flow patterns. Overall, the scattering to higher harmonics remains on a moderate level, even at very high excitation levels for the Helmholtz resonator configurations. For the quarter-wave resonator, the scattering is more pronounced and contributes perceptibly to the response at high excitation amplitudes.