## 2 Material and Methods
### Analytical Example and Finite Element Method(FEM) Simulation  
The value range of \(\theta\) of the Archimedes spiral is selected as \([0,\ 3.8]\ast 2\pi\) in the polar coordinate, and the arc length is obtained as \(L_{\mathrm{as}}=205.89\) mm. The geometrical parameters of the example HR corresponding to Figure 1b were listed as follows: \(L_{f}\)= 2 mm, \(L_{\mathrm{s}}=212.60\) mm, \(S_{\mathrm{n}}=4\) mm\({}^{2}\) and \(V=\pi\times(16\) mm\({}^{2}\times 12\) mm. All the walls of the duct, spiral neck and chamber are assumed to be rigid. The fixed fluid is selected as air and the test solution is selected as pure water as an example, which means that \(\rho_{f}\ =\ 1.29\) Kg/m\({}^{3}\), \(c_{f}\ =340\) m/s. For comparison, a conventional HR which has the same cavity and the same height of neck as the proposed HR is designed. Spectra of the transmission loss for the two types of HR are shown in Figure 3 and were obtained by Equation (6). As shown in Figure 3, the resonant frequencies are 328.47 and 2151.03 Hz, which means that the proposed HR can significantly lower the resonant frequency. In addition, the proposed HR narrows the resonance bandwidth compared with the conventional HR. This was mentioned in previous research [9] as the disadvantage of increasing neck length when the HR was applied for noise attenuation. In our research, however, only the resonant frequency was of concern, changing depending on the analyte concentration. The effect on bandwidth is not investigated as a key point in this paper. Even the sharp bandwidth can give a positive effect on reading the resonant frequency since it improves frequency resolution. To investigate the characteristics of the proposed HR more intuitively, a finite element model was established utilizing LMS virtual.lab 10-SL1 (@Siemens PLM, Munich, German) for the same model whose parameters were shown previously, and corresponding acoustic finite element analysis was carried out. In the finite element model, only the fluid domain was concerned. This means that all the walls are considered as rigid. There are 74,574 elements (tetrahedron elements) in the model and all elements were set as acoustic elements. The established acoustic finite element model with which the acoustic mode analysis was carried out is shown in Figure 4a. The pressure distribution of the 1 st mode is shown in Figure 4b, and the obtained resonant frequency is 325.08 Hz. It shows that under that mode, the acoustic pressures are concentrated on the HR, and it accounts for the mechanism of transmission loss of the duct with HR. The relative error between the analytical result using Equation (4) and the Finite Element Method (FEM) result is 1.03%, which means that the two resonant frequencies are quite close and the reasonability of the analytical derivation can be verified. The error of 1.03% may come from the error of actual spiral neck length in the analytical example and FEM simulation. Then, based on the mode analysis results, acoustic response analysis was conducted by defining unit velocity with a frequency range of 310-350 Hz in the incident end. The other duct end was defined as an anechoic end. Acoustic pressure levels were acquired in the input and output end of the duct, and then the transmission loss could be obtained using Equation (5). Figure 5 shows the spectrum of transmission loss for the duct-HR system filled with 0 wt.% (water) and 5 wt.% glucose solution. It  
Figure 3: Resonant frequency of the proposed HR and conventional HR. reveals that the peak of the transmission loss (indicating the resonant frequency of the proposed HR) shifts to a higher frequency with higher concentration. In Figure 5, two negative peaks (valleys) can be observed, and the valley frequency also shifts to a higher frequency. They are expected to occur from the other acoustic mode of the duct-HR system. In this paper, we are only concerned with the positive peaks which can be measured in the following experiment.