## 5 Acoustic Simulations  
The numerical simulations were carried out to highlight the differences in the acoustic response under the existing conditions and the original form of the Roman theatre of Gubbio. For the original form, four specific scenarios were simulated to evaluate the individual contributions of the coffered ceiling, the crowning porch, and the addition of both to the main architecture consisting of the seating area and the scenic building. All five scenarios are summarised and described below. * Scenario **A**: existing conditions. * Scenario **B**: original reconstruction provided with seating area and scenic building. * Scenario **C**: original reconstruction provided with only the coffered roof added over the stage floor. * Scenario **D**: original reconstruction provided with only the crowning porch added. * Scenario **E**: original reconstruction provided with addition of roof and porch. All five scenarios were simulated without and with an audience at 100% occupancy [21]. The geometric features of the two digital models are summarised in Table 2.  
The standard procedure for performing acoustic simulations begins with calibrating the model based on measured values [22]. If acoustic measurements are not available, the absorption and scattering coefficients are taken from the literature and are based on previous experience with acoustic simulations performed at other ancient theatres [23]. The scattering coefficients are based on the properties of the material of the external surface. The materials corresponding to the existing conditions are considered to be more non-uniform than the original marbles from antiquity. In order to take into account the possibility of different degrees of deterioration of the stone (e.g., grass/soil covering the blocks in _opus quadrature_, carved stone in _opus siliceum_ while exposed to meteorological conditions), the simulation of the existing conditions was conducted based on two different scattering coefficients at medium frequencies (500-1000 Hz): \(s=0.3\) and \(s=0.5\)[2, 24]. Figure 4 shows the current state of the stone in the Roman theatre of Gubbio. Table 3 summarises the absorption and scattering coefficients used for the simulations of the existing conditions, while Table 4 summarises the absorption and scattering coefficients used for the simulations related to the original shape of the theatre. \begin{table}
\begin{tabular}{c c c} \hline
**Description** & **EC** & **OR** \\ \hline Total number of surfaces & 18,286 & 21,054 \\ Total surface area (m\({}^{2}\)) & 16,353 & 19,192 \\ \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the two digital models used for the acoustic simulations: existing conditions (EC) and original reconstruction with all architectural elements (OR). It must be noted that the results of the simulations related to the Roman theatre of Gubbio are not to be considered as absolute values but should be taken as a reference to understand the acoustics of the existing condition compared to the reconstructed shape, with limitations represented by the variability related to the properties of the materials. In addition, the contribution of each architectural element to the overall acoustics was evaluated in the five different scenarios. The scattering coefficients are equally as important as the absorption coefficients [25], as they allow the full calibration of the digital models to compensate for the geometric simplifications applied to obtain the results with the acoustic software. The calibration process consists of a loop reiterated several times, based on the adjustment of the absorption and scattering coefficients associated with the digital entities of the model [26]. In general, the tuning process is considered valid if the difference between the simulated and reference values does not exceed 5% across all the frequency bands. However, a negligible drift deviation is attributed to physical factors, such as air absorption, and not to the quantity of surfaces that make up the geometric model.