Theoretical and experimental study of coupled Helmholtz resonators

In the present work we study single and coupled Helmholtz resonators (HR) using theoretical and experimental methods. On the theoretical side, we present two analytical approaches to obtain the resonant frequencies of some HR configurations: an elementary lumped-parameter model analogous to a system of mechanical oscillators, valid only in the long-wavelength approximation; and a more sophisticated impedance-function technique that takes into account the continuous and distributed nature of the system, and allows to obtain higher resonant frequencies under the assumption of plane-wave propagation, recovering the previous model in the long-wavelength limit. We then experimentally determine the resonant frequencies of the proposed configurations by directly recording with a microphone the sound pressure inside the resonators when excited with white noise. We also characterize the absorption properties of the resonators using a custom-built impedance tube, and find excellent agreement between the directly measured resonance frequencies and the frequencies corresponding to the absorption maxima. With appropriate effective length corrections for the necks and coupling connectors, both theoretical methods reproduce the experimentally found frequencies with relative errors below 5%. The project is suitable for undergraduate students and may appeal to those interested in acoustics, requiring only basic theoretical tools and inexpensive or readily available laboratory materials.