## 6 Theory, Methodology, and Calculation  
### Acoustic Parameters through Digital Simulations  
The acoustic simulations have been performed with Ramsete 3.13 software [13] by placing the omnidirectional sound sources along the axes of the horse race runway. The virtual microphones have been uniformly distributed across the audience area at 1.6 m height from the steps, for a number equal to 331 inside the Panathenaic Stadium, 385 inside the Circus Maximum, and 259 microphones inside the Stadium of Domitian [14; 15]. The simulations have been carried out without and with a full audience capacity, as shown in the results of Figure 5, by considering the main acoustic parameters as defined in ISO 3382 [16]. These results are obtained as an average of 10 times repetitions of the simulations in order to ascertain the degree of variance due to the repeatability of digital computations.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Materials**} & \multirow{2}{*}{**Area (m\({}^{2}\))**} & \multicolumn{4}{c}{**Octave Band Centre Frequency (Hz)—Absorption**} & \multicolumn{2}{c}{**Scattering**} \\ \cline{3-8}  & & **125** & **250** & **500** & **1000** & **2000** & **4000** & **(@500–1000 Hz)** \\ \hline \multirow{4}{*}{Stone} & 36,299 (PT) & \multirow{4}{*}{0.01} & \multirow{4}{*}{0.01} & \multirow{4}{*}{0.01} & \multirow{4}{*}{0.01} & \multirow{4}{*}{0.01} & \multirow{4}{*}{0.01} & \multirow{4}{*}{0.01} & \multirow{4}{*}{0.01} \\  & 127,958 (CM) & & & & & & & \\  & 32,644 (SD) & & & & & & & \\  & \(-\)(PT) & & & & & & & \\ Solid wood & 11,229 (CM) & 0.08 & 0.20 & 0.10 & 0.05 & 0.03 & 0.02 & 0.08 \\  & \(-\)(SD) & & & & & & & \\  & 9189 (PT) & & & & & & & \\ Sand & 94,145 (CM) & 0.08 & 0.10 & 0.20 & 0.25 & 0.30 & 0.35 & 0.05 \\  & 20,825 (SD) & & & & & & & \\  & 28,220 (PT) & & & & & & & \\ Audience & 57,089 (CM) & 0.51 & 0.64 & 0.75 & 0.80 & 0.82 & 0.83 & 0.40 \\  & 16,896 (SD) & & & & & & & \\  & 40,830 (PT) & & & & & & & \\ Openings & 118,832 (CM) & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 0.01 \\  & 34,871 (SD) & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Surface, absorption, and scattering coefficients applied to the surfaces of ancient _stadia_: Panathenaic Stadium (PT), Circus Maximum (CM), and Stadium of Domitian (SD).  
Figure 5a shows that the highest EDT values are found in the Circus Maximum, to be around 5 s in unoccupied conditions, while the results in the Panathenaic Stadium and in the Stadium of Domitian are very similar to each other, fluctuating around 3.5 s for the same scenario. When fully occupied, the EDT values are shifted by 1.2 s downwards for the Circus Maximum, to be equal to 3.5 s, while the EDT values relative to the Stadium of Domitian and Panathenaic Stadium are around 2.3 s and 1.8 s, respectively [17].  
In terms of reverberation, Figure 5b shows that the T20 values in the Circus Maximum reach 6.7 s when unoccupied, while a differentiation of 0.8 s is evident between the Panathenaic Stadium and the Stadium of Domitian, indicating that the T20 values are more reverberant in the latter stadium, to be around 3.9 s. When fully occupied, the T20 values in the Circus Maximum are lower and found around 4.3 s. The presence of audience results in more absorbtion in the Panathenaic Stadium than the Stadium of Domitian, with a downward shift of 2.3 s related to the first case study and equal to 1 s related to the latest one [18]. This is due to the quantity of surface area left to stone that is more in the Stadium of Domitian, as it is applied to the parapet (_balteum_) of the first and second _manianum_, and to the area of corridors that includes also the walkway located at the end of the steps. In the Panathenaic Stadium, the surface area left to stone is only the parapet at the level of ground and only one corridor separating the two sectors of steps [19].  
As anticipated previously, the acoustics of these three _stadia_ are very good and comparable to a large Roman theatre like Verona [20], considering that their room volumes are consistent.