# Coherent Image Source Modeling of Sound Fields in Long Spaces with a Sound-Absorbing Ceiling +
## 2 Theoretical Method
)\) is the first Hankel function with zero-th order. In Equation (5), the ray field from single reflection on the reflective boundaries, \(P_{-1,0}\) for example, can be further evaluated as [13; 17]  
\[P_{-1,0}=Q_{ref}(S_{-1,0},R|\text{GB})\cdot\frac{e^{j\mathbf{k}R_{-1,0}}}{4 \pi R_{-1,0}} \tag{6}\]  
where \(Q_{ref}(S_{-1,0},R|\text{GB})\) represents the single reflection coefficient on the reflective ground boundary (GB) and is evaluated as [13; 17]  
\[Q_{ref}(S_{-1,0},R|\text{GB})=V_{g}(\theta_{-1,0})+[1-V_{g}(\theta_{-1,0})]F (w_{n}), \tag{7}\]  
in which  
\[F(w_{n})=1+j\sqrt{\pi}\cdot w_{n}\cdot g(w_{n}), \tag{8}\]  
\[w_{n}=\sqrt{kR_{-1,0}}\cdot\frac{1+j}{2}\cdot(\cos\theta_{-1,0}+\beta_{g}), \text{ and } \tag{9}\]  
\[g(w_{n})=e^{-w_{n}^{2}}\text{erfc}(jw_{n}). \tag{10}\]  
Further analytical approximation of g (\(w_{n}\)) is available in Ref. [18]. It was shown that the wave front shape before and after each reflection on the reflective boundaries can almost remain the same [14; 19]. This suggests that single reflection coefficients \(Q_{ref}\) shall be weakly dependent on \(\theta\) and be almost uniform for different spatial parts of incident wave fronts of any shapes [14]. Accordingly, during the ray propagation from \(S_{n,m}\) to receiver in Figure 1, the evaluation of each reflection upon one reflective boundary (or each \"transmission\" through it or its images) can be approximated by once-weighting the ray field with the corresponding single reflection coefficient \(Q_{ref}\) on this boundary [14]. Thus, after the ray field being weighted for \(n_{g}\), \(m_{l}\), and \(m_{r}\) times due to \"transmission\" through the reflective ground, left wall (LW) and right wall (RW), and their images, the evaluation of \(P_{n,m}\) can be simplified by  
\[ \begin{array}{l}P_{n,m}\approx\left[Q_{ref}(S_{n,m},R\left|\text{GB}\right) \right]^{n_{g}}\cdot\left[Q_{ref}(S_{n,m},R\left|\text{LW}\right)\right]^{m_{l}} \cdot\left[Q_{ref}(S_{n,m},R\left|\text{RW}\right)\right]^{m_{r}}\\ \cdot\frac{ik}{8\pi}\int_{-\frac{ik}{2}+j\infty}^{\frac{k}{2}-j\infty}\left[V_{ c}(\theta)\right]^{n_{c}}\sin\theta\cdot H_{0}^{1}(kr\sin\theta)\ e^{ikR_{n,m}\cos\theta\cos\theta_{n,m}}d\theta,\end{array}  \tag{11}\]  
where the integral involves only the reflection coefficient on the absorptive ceiling boundary (CB) and can be further evaluated through the second order approximation provided by [PERSON] [15] to yield  
\[ \begin{array}{l}P_{n,m}\approx\left[Q_{ref}(S_{n,m},R\left|\text{GB}\right) \right]^{n_{g}}\cdot\left[Q_{ref}(S_{n,m},R\left|\text{LW}\right)\right]^{m_{l }}\cdot\left[Q_{ref}(S_{n,m},R\left|\text{RW}\right)\right]^{m_{r}}\\ \cdot\left\{V_{t}(\theta_{n,m}\left|\text{CB},n_{c})-\frac{\left[V_{t}^{\prime }(\theta_{n,m}\left|\text{CB},n_{c})\cot\theta_{n,m}+V^{\prime\prime}(\theta_{ n,m}\left|\text{CB},n_{c}\right|)\right]}{2K_{n,m}}\right\}\cdot\frac{e^{ikR_{n,m}}}{2 \pi R_{n,m}},\end{array}  \tag{12}\]  
where \(V_{t}(\theta_{n,m}\left|\text{CB},\,n_{c})=[V_{c}(\theta_{n,m})]^{n_{c}}\), and \(V^{\prime}{}_{t}(\theta_{n,m}\left|\text{CB},n_{c}\right. )\) and \(V^{\prime\prime}{}_{t}(\theta_{n,m}\left|\text{CB},n_{c}\right. )\) are the first and second derivatives of \(V_{t}(\theta_{n,m}\left|\text{CB},n_{c}\right. )\) at \(\theta_{n,m}\), respectively. This equation may be rewritten as an image source model form as  
\[P_{n,m}=Q_{n,m}\frac{e^{ikR_{n,m}}}{4\pi R_{n,m}}, \tag{13}\]  
where \(Q_{n,m}\) represents a combined reflection coefficient corresponding to the ray with reflection order (\(n\),\(m\)) as  
\[Q_{n,m}=\left[Q_{ref}(S_{n,m},R\left|\text{GB}\right)\right]^{n_{g}}\cdot \left[Q_{ref}(S_{n,m},R\left|\text{LW}\right)\right]^{m_{l}}\cdot\left[Q_{ref} (S_{n,m},R\left|\text{RW}\right)\right]^{m_{r}}\cdot Q_{abs}(S_{n,m},R\left| \text{CB},n_{c}) \tag{14}\]  
in which \(Q_{abs}\) (\(S_{n,m}\), \(R\left|\text{ CB},n_{c}\right.