# Coherent Image Source Modeling of Sound Fields in Long Spaces with a Sound-Absorbing Ceiling +
## 3 Results and Discussion
From Figure 2, the absorption coefficients of the ceiling, ground, and walls at a frequency of 1000 Hz are 0.98, 0.02, and 0.04, respectively. Figure 4 shows remarkably good agreement between the proposed model and the wave theory, even at receiver locations far away from the source compared to the space height and width. However, large prediction differences can be found between the method of [PERSON] and [PERSON] and the reference method in this situation. In Figure 4, prediction error from the proposed method increases at a longer source/receiver distance. The reason may be that, when the receiver moves farther away from the source compared to the space cross section dimensions, high-order reflection rays provide relatively higher contributions in the receiver total sound field. In the proposed model, reflection ray field is evaluated through Equations (13) and (14) by approximating each reflection at reflective boundaries as one single reflection. This may accumulate larger errors for higher-order reflection rays. Figure 3: Comparison of predictions on sound pressure level (SPL) spectrum in a long space with 20 m wide and 5 m high in numerical simulations. The source is located at (6 m, 0 m, 1 m), and the receiver is located at (6 m, 50 m, 1 m). The solid line represents the results with the proposed method, the solid circles represent the results with a dot-dash line are those with the wave theory that is considered a benchmark, and the dash line is the result with the method of [PERSON] ([PERSON]’s method). Figure 5 presents the SPL predictions at frequency of 1000 Hz versus the receiver location along the tunnel width direction. The source is located at (6 m, 0 m, 1 m) and the receiver is located at (\(x_{r}\), 50 m, 1 m) with \(x_{r}\) moving from 1 m to 19 m. The results show excellent prediction agreement between the proposed model and the wave theory, even at receiver locations close to boundary interaction corners compared to the wavelength. In Figure 5, large discrepancies remain between the method of [PERSON] and [PERSON], and the reference method. Predictions are also compared on the SPL at 1000 Hz versus the receiver location along the tunnel height direction, with the source at (6 m, 0 m, 1 m) and the receiver at (6 m, 50 m, \(z_{r}\)) with \(z_{r}\) moving from 0.125 m to 4.875 m. The corresponding results are presented in Figure 6. It is shown that results from the proposed model almost overlap those from the wave theory, not only at receiver locations close to the reflective ground but also at those in the vicinity of the absorbent ceiling compared to the wavelength. Predictions with the method of [PERSON] and [PERSON] still bias much from the benchmark results in this situation. The computational time in this simulation set is also recorded for comparison. In Matlab 2010b on the same computer as mentioned above, an evaluation of the proposed model and the method of [PERSON] and [PERSON] takes about 1.8 s and 1.7 s for results at each receiver location in Figure 4 to Figure 6, respectively, while the corresponding calculation with the wave theory takes over 80 s. These show that the proposed coherent image source model can accurately predict the sound fields in long rectangular spaces with an absorbent ceiling, while its computational load stays at a same level with the existing models [3; 9; 12; 13]. Figure 4: Comparison of predictions on sound pressure level (SPL) at frequency of 1000 Hz vs. the receiver location along the y-direction. The source is located at (6 m, 0 m, 1 m) and the receiver is located at (6 m, yr, 1 m) with yr moving from 5 m to 200 m. The solid line represents the results with the proposed method, the solid circles with a dot-dash line are those with the wave theory that is considered the benchmark, and the dash line is the result with the method of [PERSON] ([PERSON]’s method). A scale-model experiment was carried out to further verify the predictions.