Improved calculation of resonant frequencies of Helmholtz resonators

Abstract

The classical formula for calculation of resonant frequencies of Helmholtz resonators is derived under the assumptions that all mass significant for oscillation of a resonator is concentrated in the neck of the resonator and that the spring constant is given by the volume of the resonator. In this paper an extended theory is described which includes the effects of motion of mass particles inside the resonator. It is shown from this theory that the resonant frequency also depends on the shape of the vessel of the resonator and a general formula involving shape of the vessel is derived. Formulae for particular fundamental shapes —a sphere, prism, frustrum of a cone, toroid, cylinder, λ 4 resonator and cone—are presented. The theory is valid not only for Helmholtz resonators but, surprisingly enough, for such different shapes as a closed branched tube or λ 4 resonator. Thus the theory links two phenomena, resonances of Helmholtz resonators and resonances of closed branched tubes, treated separately until now, and it makes possible prediction of resonant frequencies of some cavities for which the method of calculation has been unknown until now. The theory has been verified in a series of experiments, the results of which are also stated in this paper.