Calculation of the Resonant Frequencies of the Throat Part of the Helmholtz Resonator

The calculation of the resonant frequencies of the Helmholtz resonator neck for the purpose of further evaluating the relationship between the resonant frequencies, obtained experimentally with the resonant frequencies of one of the main elements of the resonator to create broadband acoustic systems or metamaterials is presented. Recently, the issues concerning Helmholtz's resonators are of increasing interest in the scientific world. They are actively used to solve various problems in acoustics and find more and more applications - from use as mufflers, for use in the composition of metamaterials. As a simplification, it is assumed that the Helmholtz resonator has only one resonant frequency. In practice, the presence of a number of additional resonances was discovered. In this paper, it is proposed to approach the question of finding additional resonant frequencies on the other hand and consider the cavity as a system consisting of two more simple elements, to go by solving two more simple problems and to bring the results together, to reconcile them. The aim of the work is to calculate the resonant frequencies of the Helmholtz resonator neck. The throat parameters are chosen equal to the throat parameters of one of the resonators investigated in experiment. Using the MathCad software environment, the necessary calculations to obtain the values of resonant frequencies in the range from 0 to 4200 Hz were made. The 9 resonant frequencies in the range from 99 to 4060 Hz was obtained experimentally. As a result of the calculation, numerical values are obtained for the first sixty four resonant frequencies. 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 in 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). At a frequency of 3978 Hz, resonance was observed inside the flask experimentally; in the throat region, resonance was absent. At the frequencies 3037Hz, 3568Hz and 4060Hz, the resonance in the neck was poorly expressed. It can be concluded that the resonant frequencies of the throat, exceeding the value of 3000 Hz, will not have a significant effect on the resonance values of the entire resonator, i.e. for these values of frequency, resonance in the throat part was not observed experimentally. Taking into account the amplitudes of oscillations inside the neck will help reduce the number of frequencies that are interesting for further calculations and select among them the frequencies that will have the greatest influence on the value of the resonance frequencies of the entire resonator, that is, they will have the greatest amplitudes of fluctuations within the neck. Therefore, the purpose of further research is to calculate the amplitudes of oscillations that correspond to the received resonance frequencies. Ref. 12, fig. 3, tabl. 3.