## 3 Experimental Setup
At first glance, this seems not to be in agreement with Equation (4). It is worth noticing that the Euler-Bernoulli equation with viscous drag force term in Equation (3) is valid only when the gas is supposed to be in the viscous regime. Lowering the pressure, the gas can enter the molecular region, where damping mechanisms are caused by independent collisions of non-interacting molecules with the vibrating QTF prong. As a result, at pressures <10 Torr, the additive mass cannot be supposed proportional to the air density and a deviation from the linearity can also be supposed for the QTF resonance frequency. With respect to the bare QTF, the coupling with a pair of resonator tubes causes a shift of spectrophone resonance frequency. Moreover, the frequency shift increases with the pressure, although it is almost constant at pressures lower than 200 Torr. This can be explained by considering that the tubes and the QTF interact with each other through the medium in SSH at higher pressures. Thus, at low pressure, the QTF-tubes coupling is reduced and the QTF can be considered almost isolated from the tubes. For any pressure values, the frequency shift is larger when \(d=100\)\(\upmu\)m with respect to \(d=200\)\(\upmu\)m. This can be explained observing that the QTF-tube distance also affects the resonator's acoustic interaction, thus leading to a greater frequency shift for a shorter QTF-tube distance. From the linear fits, the intercept value represents the fundamental vibrational mode \(f^{\prime}_{0}\) in vacuum whereas the slope is related to the additive mass, as described in Equation (4). Indeed, as discussed in the previous section, the additive mass per unit of length \(u=u_{0}\cdot P\) is proportional to the gas pressure, where \(u_{0}=kM/R\Theta\) with \(k\) in m\({}^{2}\) units. \(u_{0}\) represents the additive mass per unit of length and pressure, and can be retrieved from the slope \(m\) of the linear fit as \(m=u_{0}f_{0}/(2\rho A)\) where \(\rho=2650\) kg/m\({}^{3}\) and the prong section \(A=T_{2}\cdot w\). The results are listed in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Bare QTF** & \
\begin{tabular}{c} **Spectrophone** \\ \(d\) **= 100** \(\upmu\)m** \\ \end{tabular} &
\begin{tabular}{c} **Spectrophone** \\ \(d\) **= 200** \(\upmu\)m** \\ \end{tabular} \\ \hline \(f_{0}\) (Hz) & 12,455.01 & 12,455.05 & 12,455.11 \\ \hline \(m\) (Hz/Torr) & \(1.31\times 10^{-3}\) & \(1.19\times 10^{-3}\) & \(1.10\times 10^{-3}\) \\ \hline \(u_{0}\) (kg/(m \(\times\) Torr)) & \(1.95\times 10^{-10}\) & \(1.77\times 10^{-10}\) & \(1.64\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters retrieved from the linear fits shown in Figure 5. \(f^{\prime}_{0}\) is the intercept of the linear fit and represents the fundamental vibrational mode in vacuum; \(m\) is the slope of the linear fit; \(u_{0}\) is the additive mass per unit of length and pressure extracted by the slope. Figure 5: Resonance frequencies (datapoints) as a function of the pressure in the three configurations. The solid lines are the best linear fits. The bare T-QTF shows a slightly higher value of \(u_{0}\) with respect to the one obtained when it is coupled with tubes. This could be related to the fact that the presence of the tubes reduces the number of molecules surrounding the QTF which results in a decreasing of the inertia to the prong oscillation. The additional inertia of the vibrating prong due to the effect of the surrounding medium was modelled as an additive mass. Indeed, with respect to the bare QTF, the reductions in the additive mass results were 8% and 16% for QTF-tube distance \(d=100\)\(\upmu\)m and \(d=200\)\(\upmu\)m, respectively. The quality factor values are extracted by fitting the resonance curves with Equation (10). The \(Q\)-factor values (datapoints) are plotted as a function of the air pressure in Figure 6, for the three investigated configurations.