## 5 Discussion  
In the results section, the FEM results were compared with the results of [PERSON] et al. 's [33] theoretical formulation and with the outcome of the lumped element approximation in cases of single-neck and multi-neck Helmholtz resonators. Various cases of Helmholtz resonators were examined: with cylindrical or spherical bodies, with unflanged or flanged necks of various dimensions and with various combinations of the above. A comparison of the results for the single-neck Helmholtz resonators between the FEM model and lumped-element approximation show very little deviation with an error of calculation less than 0.1% (Table 1). These findings are in line with research in the literature that demonstrate the effectiveness of the FEM in determining a (single-neck) Helmholtz resonator's resonance frequency (e.g., [47]). The above is an indication that our single-neck models are structured correctly, behave as expected and are a good starting base for modeling the multi-neck Helmholtz resonators. For the multi-neck Helmholtz resonators, the overall results of the resonance frequency calculated with the FEM, the [PERSON] et al. [33] formula and lumped element approximation (multi-neck) are presented in Table 1. The error of calculation between the results of the FEM and [PERSON] et al. 's [33] formula, as well as the FEM and lumped element \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Resonant** & **Resonant** & **Resonant** & **Error of** & **Error of** \\  & **Frequency** & **Frequency** & **Frequency** & **Calculation (\%)** & **Calculation (\%)** \\  & **LEA (Hz)** & **Langfeldt (Hz)** & **FEM (Hz)** & **FEM-Langfeldt** & **FEM-LEA** \\ \hline \multicolumn{6}{c}{**Single-neck models**} \\ \hline Spherical & 123.8 & - & 123.7 & - & 0.1 \\ Cylindrical & 144.0 & - & 144.0 & - & 0.0 \\ \hline \multicolumn{6}{c}{**Multi-neck models**} \\ \hline Spherical \#1 & 249.4 & 249.4 & 249.7 & 0.1 & 0.1 \\ Cylindrical \#1 & 203.6 & 203.6 & 204.3 & 0.3 & 0.3 \\ Spherical \#2 & 181.8 & 197.1 & 199.3 & 1.1 & 8.8 \\ Cylindrical \#2 & 148.4 & 161.0 & 161.3 & 0.2 & 8.0 \\ Cylindrical \#3 & 148.4 & 245.7 & 248.5 & 1.1 & 40.3 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the results of the resonant frequency calculated with the lumped element approximation, with the Langfeldt formula and with FEM. Error of calculation is also presented. Figure 14: Solid line: sound pressure level in the body of the Helmholtz resonator calculated with FEM. Dotted line: resonance frequency calculated with lumped element approximation (multi-neck). Dashed line: resonance frequency calculated with [PERSON] et al.’s [33] formula. approximation (multi-neck) is also presented. It is obvious that the addition of an extra neck causes the resonance frequency to increase, something that has also been observed in related studies [33; 34]. A comparison of the results between the FEM and [PERSON] et al. 's [33] formula shows very small differences, which, in each case, has an error of calculation less than 1.1%. In contrast, a comparison of the results between the FEM and the lumped element approximation (multi-neck) shows significant differences which can reach up to a 40.3% error of calculation. This demonstrates that the lumped element approximation for multi-neck Helmholtz resonators is not suitable for the estimation of the resonance frequencies. The same was also pointed out by [PERSON] et al. [34] and [PERSON] et al. [33] that accounting for the other necks in the Helmholtz resonator by merely adding the aggregate surface area of the additional necks to that of the principal neck could not provide accurate results. Therefore, the contribution of this research can be focused on two important areas. First, that the lumped element approximation for multi-neck Helmholtz resonators is not suitable for the estimation of the resonance frequencies in cases of multi-neck Helmholtz resonators as, in some cases, it can have a very large deviation.