Calculation of the offset amplitudes in the Helmholtz resonators’ throat part at the resonance frequencies

Abstract

The purpose of the work is to analyze the amplitudes of the oscillations in the Helmholtz resonators’ throat at the resonance frequencies. This type of resonators is actively used to solve various problems in acoustics: is widely used as the basic element in the acoustic metamaterial structure (in particular, focusing metamaterials), as the basic element of muffler etc. So the relevance of the research related to Helmholtz resonators is increasing. It is assumed as a simplification, that the Helmholtz resonator has only one resonant frequency, but the great number of additional resonances was detected in practice. The new approach to the question of finding additional resonant frequencies is shown in this paper. Here is the proposition to consider the cavity as a system consisting of two more simple elements, to go by solving two more simple tasks and to bring the results together, to reconcile them. The calculation of the resonant frequencies and the offset amplitudes of the Helmholtz resonators’ throat are given for a frequency range from 0 to 4409 Hz. These results were used for further evaluating the relationship between the resonant frequencies, obtained experimentally (for Helmholtz resonator) with the calculated resonant frequencies of one of the main elements of the resonator. The existence of the additional resonances can help to improve old designs or to create new designs of the broadband acoustic systems or acoustic metamaterials based on Helmholtz resonators. Using the Matlab software environment, the necessary calculations to obtain the values of resonant frequencies and the values of the offset amplitudes in the range from 0 to 4200 Hz were made. The throat parameters were chosen equal to the throat parameters of one of the resonators investigated in the experiment. As a result of the calculation, numerical values of the first sixty seven resonance frequencies and the values of the oscillation amplitudes on these frequencies are obtained. Some of these resonances will not have a significant effect on the resonances of the system, due to the smallness of their amplitudes of oscillations. This may explain the difference of the number of resonances obtained as a result of the calculation and experimentally (the smaller the amplitude of the oscillations, the more difficult to fix the resonance experimentally). The calculated results were compared with the results obtained experimentally (9 resonant frequencies in the range from 99 to 4060 Hz were obtained experimentally). The analysis of the results is presented in the article. According to the results of the calculation and of the experiment, the significant resonances won’t be observed in the throat part of the resonator at some resonant frequencies of the system (Helmholtz resonator). According to the results of the experiment the resonance of the bulb was detected on such frequencies. To take into account the influence of the bulb on the value of the resonance frequencies of the Helmholtz resonator, it is necessary to analyze the resonance frequencies of the bulb of the selected resonator. Therefore, the purpose of further research will be to calculate and estimate the resonant frequencies of the Helmholtz resonator bulb. Ref. 13, fig.3, tabl. 4.