Design relation and end correction formula for multi-orifice intruded Helmholtz resonators

Helmholtz resonators (HRs) are used to control low frequency noise in cavities. One of the ways to reduce the natural frequency of a resonator without changing its volume is to introduce an intrusion. Similarly the introduction of multiple orifices can increase the natural frequency without changing the resonator volume. These features provide an ability to accomodate slight changes in the cavity/enclosure modes during the design process. However, one has to rely on extensive 3D finite element or boundary element simulations to predict the resonator charachetristics with the introduction of these features. Hence, the primary objective is to develop, a design relation between first frequency of a single orifice intruded resonator and a multi-orifice intruded resonator at various intruded lengths. In developing this design relation, the total cross-sectional area of the resonator with multiple orifices is kept the same as that of the single orifice resonator. It is shown that the design relation is independent of the shape/size of the orifices as well as the resonator cavity volume. Limited experiments have been carried out to validate the simulation. Simple lumped parameter models are available for single orifice Helmholtz resonators. No such generalized models are available for multi-orifice intruded HRs. A new end correction formula, for the orifice lengths of multi-orifice intruded resonators, has been derived from the design relation. The end correction formula can be used to calculate the reactance of multi-orifice intruded HRs analytically. This simplifies the coupling of HR to the cavity to a greater extent, as modelling several resonators along with the cavity is computationally intensive due to the requirement of fine meshing around the resonators.