# 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
#### 5.3.1 Results and Discussion  
Figure 13 portrays comparisons of direct sound waveforms at R1 (\(x\), \(y\)) = (15.1, 1.25) for the reference solution and the proposed explicit TD-FEM and the standard implicit TD-FEM for Cond. 1. Results for Cond. 2 were omitted because similar results were obtained. The proposed explicit TD-FEM shows good agreement with the reference solution, whereas the standard implicit method shows a marked discrepancy with an increased sound speed because of the larger dispersion property. Figure 14 presents a comparison of impulse responses up to 1.0 s for Cond. 1 and Cond. 2. The proposed explicit TD-FEM shows much better agreement with the reference solution in the entire time range for both conditions. However, the standard implicit TD-FEM shows poor approximation capability. In the early time region, it underestimates the amplitude of reflected waves, which is also observed for reflected waves at around 0.4 s for Cond. 1. Figure 15 presents a comparison of absolute error \(e(t)\) between the proposed method and standard method for Cond. 1 and Cond. 2, where the error is the averaged value for all receivers. It is calculated at the time after arrival of the direct sound. The proposed explicit TD-FEM presents much lower errors for both conditions with magnitude below 0.2 dB. The standard implicit TD-FEM has error magnitude of approximately 0.4 dB and 0.6 dB for Cond. 1 and Cond. 2. In addition, the error level increases for more complicated Cond. 2. Regarding the advantage of using explicit TD-FEM, it can simulate sound propagation in concert halls, with approximately one-fifth less computational time than that of the implicit standard TD-FEM and one-third times the fourth-order accurate implicit TD-FEM used for reference calculation. These results suggest the high efficiency of the proposed explicit TD-FEM for acoustic simulations in complicated sound fields at kilohertz frequencies. For reference, total numbers of MVP required for the respective methods are 392,000 for the proposed explicit method, 1,995,506-2,053,979 for the standard implicit method, and 1,219,622-1,306,921 for the fourth-order implicit method.  
Figure 16 presents sound propagation in the concert hall for both conditions calculated using the proposed explicit TD-FEM. It shows again that the sound propagation is isotropic. Additionally, one can observe from the result of Cond. 2 that the effect of increased diffuseness of resulting sound fields can be gained by virtue of the sound scattering by the rib structures. It is also apparent from the effect of the reflector that strong reflection of sound from the reflector comes at an earlier time to the audience area than reflection waves from the ceiling and back wall of the stage. Furthermore, Figure 17 shows time-integrated SPL until 0.1 s for Cond. 1 and Cond. 2. By installing the reflector, it is readily apparent that early incident sound energies on the audience area increase and that sound energies around the back wall of the stage decrease. These results demonstrate the effectiveness of the proposed explicit TD-FEM clearly as a room acoustic design tool in practical applications.  
Figure 14: Band-limited impulse responses at R1 among the reference solution, the proposed explicit TD-FEM, and the standard implicit TD-FEM for Cond. 1 (upper row) and Cond. 2 (lower row).  
Figure 15: Comparison of absolute errors in time integrated sound pressure levels between the proposed explicit TD-FEM and the standard implicit TD-FEM for Cond. 1 and Cond. 2.  
Figure 16: Sound propagation in a concert hall model for Cond. 1 (left column) and Cond. 2 (right column) at \(t\) = 10, 30 and 50 ms.