# Time Domain Room Acoustic Solver with Fourth-Order Explicit FEM Using Modified Time Integration
## 5 Numerical Experiments with Practical Sound Fields
### Sound Propagation in a Concert Hall  
A second numerical example demonstrates the performance in a more practical application. Figure 11 presents analyzed concert hall models of 296 m\({}^{2}\) surface area with two conditions. In the figure, Cond. 1 is a basic model; Cond. 2 is a model including an acoustic reflector above the stage and rib structures at a back wall in the stage for sound scattering. The reflector has 5.86 m length with 0.4 m width; the rib structure has a 0.4 m period length with 0.1 m depth and 0.2 m width. We used the simplest equivalent impedance model to address sound absorption effects at boundaries. For the seat and the back wall in the audience area, represented respectively as red and yellow lines in Figure 11, we gave \(z_{n}\) = 3.87 and \(z_{n}\) = 7.14 respectively. They correspond to Paris's statistical absorption coefficients, \(\alpha_{\text{Paris}}\), of 0.8 and 0.6, which are calculated as  
\[\alpha_{\text{Paris}}=\frac{1}{z_{\text{n}}^{2}}\bigg{(}1+z_{\text{n}}-\frac{1 }{1+z_{\text{n}}}-2\text{ln}(1+z_{\text{n}})\bigg{)}. \tag{50}\]  
Other boundaries including the reflector surfaces were set as \(z_{n}\)=71.519 corresponding to \(\alpha_{\text{Paris}}\) = 0.1. As a sound source, we used an impulse response of fourth-order Butterworth type bandpass filter with 1/3 octave band width centered at 2 kHz. The source waveform and its frequency spectrum are portrayed in Figure 12. The band-limited room impulse responses were calculated up to 2 s at 15 receivers R1-R15, as listed in Table 2. The concert hall models were discretized using four-node quadrilateral FEs with maximum edge length of 0.02 m, which corresponded to 7.65 PPW at the upper-limit frequency of 2 kHz 1/3 octave band. The discretized FEM models have 806,478 DOF and 880,796 DOF, respectively, for Cond. 1 and Cond. 2. We set \(\Delta t\) as 1/98,000 s. As for the measure of accuracy, because one important evaluation in room acoustics is conducted using room acoustical parameters such as reverberation time, clarity, and strength, the absolute error in time integrated sound pressure level might become a useful measure. It is defined as shown below.  
\[e(t)=\left|20\text{log}\bigg{(}\frac{\int_{0}^{t}p_{\text{reference}}(\tau)d \tau}{\int_{0}^{t}p_{\text{TFDFEM}}(\tau)d\tau}\bigg{)}\right|\,\left[\text{ dB}\right] \tag{51}\]  
Figure 10: Comparisons of sound pressure waveforms at a receiver (\(x\), \(y\)) = (5.3, 3.0) among the reference solution, the proposed explicit TD-FEM, and the standard implicit TD-FEM.  
Figure 11: Concert hall models of two conditions (Cond. 1 and Cond. 2) where red and yellow lines respectively represent seat zone with \(z_{n}=3.87\) and back wall with \(z_{n}=7.14\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Receiver** & **(\(x,y\)), m** & **Receiver** & **(\(x,y\)), m** \\ \hline R1 & (15.1, 1.25) & R9 & (6.3, 3.5) \\ R2 & (14.1, 1.5) & R10 & (5.3, 3.9) \\ R3 & (13.1, 1.7) & R11 & (4.3, 4.3) \\ R4 & (12.1, 2.0) & R12 & (3.3, 4.7) \\ R5 & (11.1, 2.1) & R13 & (2.3, 5.1) \\ R6 & (10.1, 2.5) & R14 & (1.3, 6.3) \\ R7 & (8.3, 2.7) & R15 & (0.8, 6.7) \\ R8 & (7.3, 3.1) & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: List for 15 receiver positions (R1–R15) in concert hall model.  
Figure 12: Source signal of fourth-order Butterworth type 1/3 octave bandpass filter with center frequency of 2 kHz: (**a**) impulse response and (**b**) frequency characteristic.  
In that equation, \(p_{\text{TDFEM}}(t)\) and \(p_{\text{reference}}(t)\) respectively denote the numerical solution and reference solution at time \(t\). For computing the reference solution, the same FE meshes were used because the theoretically estimated maximum dispersion error of reference method with these meshes is lower than 0.1%.