# Coherent Image Source Modeling of Sound Fields in Long Spaces with a Sound-Absorbing Ceiling +
## 3 Results and Discussion
One model long rectangular space was built with inner dimensions of 0.7 m width, 0.45 m height, and 10 m length for the measurements, which was scaled with 1:10 to represent a tunnel with \(W\times H=7\) m \(\times\) 4.5 m in full scale (all of the following dimensions referred to are scaled ones unless otherwise stated). Panels of 20 mm thick high-density fiberboard were used to build the model long space, and the model's inner surfaces were well finished to represent sound reflective ground and wall boundaries. A layer of 50 mm thick fiberglass was used as a liner on the top panel to represent an absorbent ceiling. To minimize the sound reflection on the two ends of the model's long space for infinite extension, liners of 200 mm thick fiberglass were applied onto those two end panels. The specific normalized admittances of the model boundaries were preliminarily measured through an impedance tube kit typed B&K 4206. Figure 5: Comparison of predictions on sound pressure level (SPL) at a frequency of 1000 Hz vs. the receiver location along the \(x\)-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 solid line represents the results with the proposed method, the solid circles are those with the wave theory that is considered as a benchmark, and the dash line is the result with the method of [PERSON] ([PERSON]’s method). A speaker driver with a tube of internal diameter of 2 cm and length of 1 m was applied to represent a point source [3]. One microphone typed B&K 4190 was used as the receiver. Sound signals were generated and collected through one B&K Pulse system 3560D. In measurements, high enough levels of white noise were generated into the model long space to ensure the steady SPL at most locations inside remained at least 15 dB higher than the background noise. In accordance with coordinates defined in Figure 1, in the experiment, the point source (the speaker tube mouth) was located at (0.35 m, 0 m, 0.2 m) and the receiver was located at (0.35 m, \(y_{r}\), 0.1 m) with \(y_{r}\) moving from 0.1 m to 8 m to investigate the SPL distribution along the space extension direction. Relative attenuation (RA) was used to present the measured and predicted results in the experiment, which is defined as subtracting the SPL at (0.35 m, 0.1 m, 0.2 m) from that at receiver. The predictions from the wave theory that are used as benchmarks in simulations were firstly compared with the experimental data. Figure 7 presents the comparison results on the RA distribution along the \(y\)-direction. The frequency of 1000 Hz was chosen without loss of generality, at which \(\beta_{c}\) was (1.2068-1.4338i), corresponding to a normal incident absorption coefficient of 0.7, while \(\beta_{g}\), \(\beta_{r}\), and \(\beta_{l}\) were (0.0272 + 0.1041i), corresponding to a normal incident absorption coefficient of 0.1. In Figure 7, reasonable agreement is shown between the wave theory predictions and the experimental data. By considering experimental uncertainty and errors such as those from the receiver locations and model tunnel dimensions in measurements, this agreement supports the reliability of the benchmark results used in numerical validations above. Predictions from the proposed method and the method of [PERSON] were compared with the experimental data, as presented in Figure 7 as well. It is shown that,  
Figure 6: Comparison of predictions on sound pressure level (SPL) at frequency of 1000 Hz vs. the receiver location along the \(z\)-direction. The source is located at (6 m, 0 m, 1 m) and the receiver is located at (6 m, 50 m, \(z_{r}\)) with \(z_{r}\) moving from 0.125 m to 4.875 m. The solid line represents the results with the proposed method, the solid circles are those with the wave theory that is considered as a benchmark, and the dash line is the result with the method of [PERSON] ([PERSON]’s method).