## 2 Analytical Model
### Instantaneous Pressure
#### 2.2.2 Reflected Pressure Fields  
The reflected pressure field results from the specular reflection of the emitted acoustic waves by both the top and the bottom walls. Still using the principle of superposition, it can be expressed as  
\[p_{ref}(\mathbf{x},t)=p^{\prime}_{ref}(\mathbf{x},t)+p^{{}^{\prime\prime}}_{ref}(\mathbf{ x},t) \tag{6}\]  
where \(p^{\prime}_{ref}\) and \(p^{\prime\prime}_{ref}\), respectively, stand for the reflected field originating from the top and the bottom wall alone. Each one of these reflected pressure fields can be constructed using a source image [39], i.e., a _virtual_ source that would mirror the actual one through the considered wall (cf. Figure 1). Of note, we here limit ourselves to primary reflections, i.e., we neglect all secondary reflections that would come from multiple and successive backscatters by the top and bottom walls (e.g., noise source \(\rightarrow\) top (resp. bottom) wall \(\rightarrow\) bottom (resp. top) wall \(\rightarrow\) observer). Accounting for these additional reflections would imply considering an infinity of virtual sources, to be located one above the other (i.e., the second virtual source would correspond to the image of the first one, once mirrored by the opposite wall). Doing so would complexify the analysis unnecessarily, considering the less critical role played by these secondary sources. Indeed, compared to their primary counterparts, these further scattered waves are likely to see their amplitude decreasing rapidly, due to their comparatively longer propagation path. This, at least, is valid for observers that are located close enough to the actual source such that they experience a significant relative difference between the direct (actual) and the scattered (virtual) propagation distances. This would not hold for observers located further away from the trailing edge (e.g., \(|x-x_{s}|\gg 2\hbar\)), and for which secondary reflections should be kept. Pressure Field Reflected by the Top Wall (Alone)  
The pressure field reflected by the sole top wall corresponds to a _virtual_ incident field that would be emitted by the source image \(x^{\prime}_{s}\) (cf. Figure 1):  
\[p^{\prime}_{ref}(\mathbf{x},t)=p_{s}\big{(}r^{\prime},\theta^{\prime},t\big{)} \tag{7}\]  
where \(r^{\prime}\) and \(\theta^{\prime}\) respectively stand for the virtual propagation distance and emission angle between the mirroring source and the observer (sketched in blue color in Figure 1). With such mirroring source located in \(x^{\prime}_{s}=\left(x^{\prime}_{s},y^{\prime}_{s}\right)\), similarly to above, the virtual propagation distance becomes  
\[r^{\prime}=\left|\mathbf{x}-\mathbf{x}^{\prime}_{s}\right|=\sqrt{\left(x-x^{\prime}_{ s}\right)^{2}+\left(y-y^{\prime}_{s}\right)^{2}} \tag{8}\]By construction, the virtual emission angle is now given by  
\[\theta^{\prime}=\frac{\pi}{2}-\left(\alpha+\beta^{\prime}\right) \tag{9}\]  
where  
\[\beta^{\prime}=Atan\left(-\frac{y-y_{s}^{\prime}}{x-x_{s}^{\prime}}\right) \tag{10}\]  
Notably, in the present case of a top wall that is parallel to the \(x\)-axis, one has  
\[x_{s}^{\prime}=x_{s} \tag{11}\]  
and  
\[y_{s}^{\prime}=2h-y_{s} \tag{12}\]  
Therefore  
\[r^{\prime}=\sqrt{\left(x-x_{s}\right)^{2}+\left(y+y_{s}-2h\right)^{2}} \tag{13}\]  
and  
\[\beta^{\prime}=Atan\left(\frac{2h-\left(y+y_{s}\right)}{x-x_{s}}\right) \tag{14}\]  
Pressure Field Reflected by the Bottom Wall (Alone)  
In a similar fashion, the pressure field reflected by the sole bottom wall corresponds to a _virtual_ incident field that would be emitted by the source image \(x_{s}^{\prime\prime}\) (cf. Figure 1):  
\[p_{ref}^{\prime\prime}(x,t)=p_{s}(r^{\prime\prime},\theta^{\prime\prime},t) \tag{15}\]  
where \(r^{\prime\prime}\) and \(\theta^{\prime\prime}\) respectively stand for the virtual propagation distance and emission angle between the mirroring source and the observer (sketched in red color in Figure 1).