# Coherent Image Source Modeling of Sound Fields in Long Spaces with a Sound-Absorbing Ceiling +
## 2 Theoretical Method  
Figure 1 shows the cross-sectional geometry of a long rectangular space with a height of \(H\) and width \(W\). For simplicity, four boundaries in this space, the ceiling, ground, and right and left walls, are assumed to be locally reactive with a uniform normalized specific admittance of \(\beta_{c}\), \(\beta_{g}\), \(\beta_{r}\), and \(\beta_{l}\), respectively. The ceiling is defined to be sound absorptive with a relatively high sound absorption coefficient, while other boundaries are sound reflective with a relatively low absorption coefficient. The space extends infinitely along the \(y\)-direction as a typical case and a point source is located at (\(x_{S}\), \(0\), \(z_{S}\)) and a receiver is located at (\(x_{r}\), \(y_{r}\), \(z_{r}\)) inside. To model the sound field, we first assume that \(kW>>1\) and \(kH>>1\) (with \(k\) for the wavenumber) so that the boundaries may be considered infinity for each sound reflection on them [13]. The total sound pressure field at receiver can be approximated as a summation of successive sound reflections on four boundaries:  
\[P_{tot}\approx\sum_{n=-\infty}^{+\infty}\sum_{m=-\infty}^{+\infty}P_{n,m} \tag{1}\]  
where \(n\), \(m=0\), \(\pm 1\), \(\pm 2\),..., and \(P_{n,m}\) represents the sound field contribution from the (\(n\), \(m\))-th order image source, in which a positive \(n\) (or \(m\)) is for an image source located above the ceiling (or rightwards from the right wall) while a negative \(n\) (or \(m\)) is for that located below the floor (or leftwards from the left wall), as shown in Figure 1. Particularly, \(P_{0,0}\) denotes the direct sound from the real source \(S_{0,0}\). Based on the assumption of \(kW>>1\)  
Figure 1: Cross-sectional geometry of a long rectangular space with height H and width W, and the image sources formed by multiple reflections on its four boundaries. and \(kH>>1\), \(P_{n,m}\) can be approximated from the plane wave expansion of a spherical wave as follows [14; 15]:  
\[P_{n,m}\approx\frac{jk}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}-j\infty}e ^{i\mathbf{k}\cdot\mathbf{R}_{n,m}}[V_{g}(\theta)]^{n_{g}}[V_{c}(\theta)]^{n_{c }}[V_{l}(\alpha)]^{m_{l}}[V_{r}(\alpha)]^{m_{r}}\sin\theta\ d\theta\ d\varphi \tag{2}\]  
where \(n_{g}\), \(n_{c}\), \(m_{l}\), and \(m_{r}\) are used to count reflection times on the ground, ceiling, and left and right walls in the path from \(S_{n,m}\) to the receiver, respectively. They can be determined from the order \((n,m)\) by  
\[ \begin{array}{c}n_{g,c}=\frac{|n|}{\tau}\mp\frac{1}{2}\text{sign}(n)\text{rem }(|n|,2)\\ m_{l,r}=\frac{|m|}{2}\mp\frac{1}{2}\text{sign}(m)\text{rem}(|m|,2)\end{array}  \tag{3}\]  
In Equation (2), \(\mathbf{R}_{n,m}\)=\((R_{n,m}\text{sin}\theta_{n,m}\text{cos}\text{i}_{n,m},R_{n,m}\text{sin} \text{i}_{n,m}\text{sin}\text{i}_{n,m},R_{n,m}\text{cos}\theta_{n,m})\) represents the distance vector from \(S_{n,m}\) to the receiver, with explicit azimuth angles \(\theta_{n,m}\) and \(\text{j}_{n,m}\). \(V_{g}(\theta)\) and \(V_{c}(\text{j})\) are the plane wave reflection coefficients on the \"infinite\" ground and ceiling with the incidence angle \(\theta\), respectively, while \(V_{l}(\alpha)\) and \(V_{r}(\alpha)\) are those on the left and right walls with the incidence angle \(\alpha=\pi/2-\theta\), respectively. These plane wave reflection coefficients can be correspondingly evaluated by [16]  
\[V_{g,c,r,l}(\zeta)=\frac{\cos\zeta-\beta_{g,c,r,l}}{\cos\zeta+\beta_{g,c,r,l}} \tag{4}\]  
Through an identical mathematical transformation similar to that from Equations (8)-(12) in Ref. [14], the evaluation of \(P_{n,m}\) in Equation (2) can be simplified as  
\[P_{n,m}\approx\frac{jk}{8\pi}\int_{-\frac{\pi}{2}+j\infty}^{\frac{\pi}{2}-j \infty}V(\theta)\sin\theta\cdot H_{0}^{1}(kr\sin\theta)\ e^{j\mathbf{k}R_{n,m} \cos\theta\cos\theta_{n,m}}d\theta \tag{5}\]  
where \(V(\theta)\) represents the term \([V_{g}(\theta)]^{n_{g}}[V_{c}(\theta)]^{n_{c}}[V_{l}(\alpha)]^{m_{l}}[V_{r}( \alpha)]^{m_{r}}\), \(r=R_{n,m}\text{sin}\theta_{n,m}\), and \(H_{0}^{1}(.