# Increasing the Insertion Loss of Sonic Crystal Noise Barriers with Helmholtz Resonators
## Appendix A  
This appendix explains why viscothermal losses do not necessarily need to be considered in the modelling in the numerical simulations in this paper. To this end, a model of viscothermal losses is presented and verified by numerical simulations that its acoustic effects are negligible compared to an equivalent lossless system. The main causes of dissipation of acoustic energy in a SCNB with HR are viscous forces and thermal conduction, which occur mainly at the viscous and thermal boundary layers. Temperature and velocity gradients between the bulk domain and the boundary surface are responsible for the losses. Based on these mechanisms, a distinction can be made between a thermal and a viscous boundary layer. While the heat gradient occurs when this boundary wall is isothermal, the velocity gradient can be explained by the fluid viscosity (\(\mu\)) and the no slip condition at the wall. While the thermal boundary layer (\(\delta_{t}\)) is a function of the fluid thermal conductivity (\(\kappa\)), the viscous boundary layer (\(\delta_{v}\)) is dependent on the fluid viscosity. According to the following equations, the boundary layers are both frequency dependent [47]:  
\[\delta_{t}=\sqrt{\frac{\kappa}{\rho_{0}C_{p}\ \pi f}}\, \tag{1}\]  
and  
\[\delta_{v}=\sqrt{\frac{\mu}{\rho_{0}\ \pi f}}\, \tag{2}\]  
where \(C_{p}\) is the fluid heat capacity at constant temperature and the fluid density is \(\rho_{0}\). Both characteristic lengths are related according to the not dimensional Prandtl number, \(\Pr=\left(\delta_{v}/\delta_{t}\right)^{2}\). In this study the fluid is air with T = 20\({}^{\circ}\) C and p = 1 atm, where thermal and viscous losses are almost equally important, having a Pr = 0.7, in contrast with water or other fluids where the role of thermal losses is smaller. Modelling viscothermal losses in acoustics can be computationally expensive as a large amount of detail is required to capture all physical effects. However, if the acoustic waves travel through slits with constant cross-section much smaller than the acoustic wavelength and larger than the thickness of the boundary layers, the viscothermal losses can be estimated by including them in the fluid in a homogenized approach for the narrow domain of the slit using the Low Reduced Frequency models [32; 48]. According to the analyzed domain of the slit, the case of this study is the neck of the Helmholtz resonator embedded in the scatter and the governing equations in that zone to estimate viscothermal losses are, the complex wavenumbers by splitting it into the viscous and thermal parts, the following wavenumbers are obtained:  
\[k_{v}{}^{2}=-\mathrm{i}2\pi f\frac{\rho_{0}}{\mu}\, \tag{3}\]  
and  
\[k_{t}{}^{2}=-\mathrm{i}2\pi f\frac{\rho_{0}C_{p}}{\kappa}. \tag{4}\]  
As can be seen in (1) and (2), the viscous and thermal layers have a greater width for long wavelengths. Therefore, if their influence is negligible at low frequencies, this will also be the case at higher frequencies. For this reason, only the case where HR is tuned to low frequency is considered here to prove that viscoelastic losses can be neglected in this work. Figure 1, shows two scatterers with embedded HR. Figure 1-left shows the HR used on the work with a neck of w = 1 cm and L = 1 cm, called HR1 and Figure 1-right, the HR2 with the same theoretical resonance frequency \(\mathrm{f}_{\mathrm{HR}}\) = 400 Hz, the same length of the neck L = 1 cm and consequently a thinner mouth (w = 0.2 cm). The IL of two 3-row SCNBs with both HR1 and HR2 are calculated numerically using the model presented in Section 2.2. It is shown that the viscothermal losses for the HR studied in this work can be neglected in the working frequency range. Figure 2 illustrates the IL in the frequencies of the HR-BG. Since the mouth of the resonator is quite large, no significant viscothermal losses occur in the neck of HR1. However, when the size of the mouth is reduced, the losses are no longer negligible and the acoustic properties of the SCNB change significantly.