# Coherent Image Source Modeling of Sound Fields in Long Spaces with a Sound-Absorbing Ceiling +
## 3 Results and Discussion  
Numerical simulations are firstly carried out to validate the proposed coherent image source model. As the classic wave theory is analytically exact in the spaces studied in this paper [16], it is used as a reference method to provide benchmark results in validations. The coherent image source method by [PERSON] and [PERSON] [13] that was widely used in previous studies [3; 9; 12] is also investigated for comparisons. Numerical implementation of the methods above stays similar to that in Refs. [13; 14], except the geometry of four boundaries in Figure 1.  
In simulations, a long rectangular space with \(W\times H=20\) m \(\times\) 5 m is considered to simulate one city road tunnel with four lanes. For simplicity, four tunnel boundaries are all assumed to be rigidly backed layers of homogeneous porous material. [PERSON]'s \"three-parameter\" approximation [14; 20] is applied to evaluate surface admittances for these boundaries, in which the boundary media parameters of flow resistivity (\(\sigma\)), porosity (\(\Omega\)), tortuosity (\(T\)), pore shape factor (\(S_{p}\)), and thickness (\(d\)) are used for evaluation. The tunnel ceiling is defined as highly sound absorptive, with \(\sigma=10\) cgs (where 1 cgs = 1 kPa s m\({}^{-2}\)), \(\Omega=1\), \(T=1\), \(S_{p}=0.25\), and \(d=0.1\) m, such as a wool layer. The ground has \(\sigma=10\) k cgs, \(\Omega=0.2\), \(T=1.4\), \(S_{p}=0.5\), and \(d=0.05\) m to represent a compact asphalt pavement layer. The right and left walls have \(\sigma=0.5\) k cgs, \(\Omega=0.1\), \(T=1\), \(S_{p}=0.3\), and \(d=0.01\) m to represent cement plaster over concrete walls. Figure 2 showsthe corresponding normal incident absorption coefficients of these four boundaries in simulations. Two sets of numerical simulations are conducted. In the first set, predictions of sound pressure level (SPL) spectrum at the receiver (6 m, 50 m, 1 m) from a point source at (6 m, 0 m, 1 m) are investigated. Predictions from the proposed method, the wave theory, and the method of [PERSON] are compared in Figure 3. It is shown that the results from the proposed method agree excellently with those of the wave theory over frequencies from 500 to 2000 Hz, with only small deviations (<1 dB) at few lower frequencies. This suggests the successful extension of the coherent image source method by [PERSON] et al. [14] for spaces enclosed by four perpendicular finite boundaries in this paper. It can also be observed from Figure 3 that the predictions with the method of [PERSON] and [PERSON] differ significantly from the benchmark results over frequencies in this situation. This indicates that the existing coherent models [3,9,12] based on the method of [PERSON] and [PERSON] can hardly be accurate in long spaces with absorbent boundaries because the assumption of spherical wave front shapes for each successive reflection is unsatisfied in this situation. All simulations are executed in Matlab 2010b on the same personal computer with a 2.4 GHz Intel Core i5-560M processor and 8 Gbytes of random access memory. Computational time records show that, for results at all the 31 frequencies in Figure 3, evaluation of the proposed model and the method of [PERSON] and [PERSON] takes 50.3 s and 49.7 s, respectively, while the corresponding execution time with the wave theory takes over 2 h. This indicates the remarkable advantage of the proposed model both at accuracy and efficiency for sound predictions in long spaces with absorbent boundaries. Figure 2: Spectra of the normal incident absorption coefficient on four boundaries of the rectangular long space in numerical simulations. In the second simulation set, predictions of the SPL distribution inside the long rectangular space are investigated. Figure 4 presents SPL predictions at frequency of 1000 Hz versus the receiver location along the tunnel extension direction. The source is located at (6 m, 0 m, 1 m) and the receiver is located at (6 m, \(y_{r}\), 1 m) with \(y_{r}\) moving from 5 m to 200 m along the space length-extending direction.