## 2 Analytical Model
### Instantaneous Pressure
#### 2.2.4 Incident, Reflected and Total Pressure Fields at the Wall  
Let us consider the specific (albeit common) situation where in situ noise measurements are achieved using a flush-mounted microphone that is placed on one of the duct walls. In this case, the observer coordinates become \(\mathbf{x}_{\mathbf{M}}^{\pm}\ =\ (x_{M},\pm h)\). For an observer that would be placed on the top wall (i.e., \(\mathbf{x}_{\mathbf{M}}^{+}\ =\ (x_{M},h)\)), the overall pressure field thus becomes  
\[p\big{(}\mathbf{x}_{\mathbf{M}}^{+}t\big{)}=p_{\rm s}(r_{+},\theta_{+},t)+p_{\rm s} \big{(}r_{+}{{}^{\prime}},\theta_{+}{{}^{\prime}},t\big{)}+p_{\rm s}(r_{+}{{} ^{\prime\prime}},\theta_{+}{{}^{\prime\prime}},t) \tag{28}\]  
with  
\[r_{+}=\sqrt{\left(x_{M}-x_{\rm s}\right)^{2}+\left(h-y_{\rm s}\right)^{2}} \qquad\theta_{+}=\frac{\pi}{2}-\left(\alpha+\beta_{+}\right)\qquad\beta_{+}= Atan\bigg{(}\frac{h-y_{\rm s}}{x_{M}-x_{\rm s}}\bigg{)} \tag{29}\]  
\[r_{+}{{}^{\prime}}=\sqrt{\left(x_{M}-x_{\rm s}\right)^{2}+\left(y_{\rm s}-h \right)^{2}}\qquad\theta_{+}{{}^{\prime}}=\frac{\pi}{2}-\left(\alpha+\beta_{+ }{{}^{\prime}}\right)\qquad\beta_{+}{{}^{\prime}}=Atan\bigg{(}\frac{h-y_{\rm s }}{x_{M}-x_{\rm s}}\bigg{)} \tag{30}\]  
\[r_{+}{{}^{\prime\prime}}=\sqrt{\left(x_{M}-x_{\rm s}\right)^{2}+\left(y_{\rm s }+3h\right)^{2}}\qquad\theta_{+}{{}^{\prime\prime}}=\frac{\pi}{2}-\left( \alpha-\beta_{+}^{\prime\prime}\right)\qquad\beta_{+}{{}^{\prime\prime}}=Atan \bigg{(}\frac{3h+y_{\rm s}}{x_{M}-x_{\rm s}}\bigg{)} \tag{31}\]  
One can here notice that \(r_{+}{{}^{\prime}}=r_{+}\) and \(\beta_{+}{{}^{\prime}}=\beta_{+}\), i.e., \(\theta_{+}{{}^{\prime}}=\theta_{+}\). Therefore, in this particular case, the acoustic field that is reflected by the top wall matches exactly the incident one. The latter will thus be doubled up by the former, due to perfectly correlated constructive interferences. On the other hand, the acoustic field that is reflected by the bottom wall will bring an additional modulation, through either constructive or destructive interferences.  
\[p\big{(}\mathbf{x}_{\mathbf{M}}^{+},t\big{)}=2p_{\rm s}(r_{+},\theta_{+},t)+p_{\rm s}(r _{+}{{}^{\prime\prime}},\theta_{+}{{}^{\prime\prime}},t) \tag{32}\]  
For an observer that would be placed on the bottom wall \(\mathbf{x}_{\mathbf{M}}^{-}=(x_{M},-h)\), the pressure becomes  
\[p\big{(}\mathbf{x}_{\mathbf{M}}^{-},t\big{)}=p_{\rm s}(r_{-},\theta_{-},t)+p_{\rm s} \big{(}r_{-}{{}^{\prime}},\theta_{-}{{}^{\prime}},t\big{)}+p_{\rm s}(r_{-}{{} ^{\prime\prime}},\theta_{-}{{}^{\prime\prime}},t) \tag{33}\]  
with  
\[r_{-}=\sqrt{\left(x_{M}-x_{\rm s}\right)^{2}+\left(h+y_{\rm s}\right)^{2}} \qquad\theta_{-}=\frac{\pi}{2}-\left(\alpha+\beta_{-}\right)\qquad\beta_{-}=- Atan\bigg{(}\frac{h+y_{\rm s}}{x_{M}-x_{\rm s}}\bigg{)} \tag{34}\]\[r_{-}{{}^{\prime}}=\sqrt{\left(x_{M}-x_{s}\right)^{2}+\left(y_{s}-3h\right)^{2}} \qquad\theta_{-}{{}^{\prime}}=\frac{\pi}{2}-\left(\alpha+\beta_{-}{{}^{\prime} }\right)\qquad\quad\beta_{-}{{}^{\prime}}=Atan\left(\frac{3h-y_{s}}{x_{M}-x_{s }}\right) \tag{35}\]  
\[r_{-}{{}^{\prime\prime}}=\sqrt{\left(x_{M}-x_{s}\right)^{2}+\left(y_{s}+h \right)^{2}}\qquad\theta_{-}{{}^{\prime\prime}}=\frac{\pi}{2}-\left(\alpha- \beta_{-}{{}^{\prime\prime}}\right)\qquad\beta_{-}{{}^{\prime\prime}}=Atan \left(\frac{h+y_{s}}{x_{M}-x_{s}}\right) \tag{36}\]  
In this case, \(r_{-}{{}^{\prime\prime}}=r_{-}\) and \(\beta_{-}{{}^{\prime\prime}}=-\beta_{-}\), i.e., \(\theta_{-}{{}^{\prime\prime}}=\theta_{-}\), that is, the acoustic field reflected by the bottom wall is the same as the incident one. Similar to what happens for its top wall counterpart, both fields will sum up, due to perfectly correlated constructive interferences. The result will then be further modulated due to the acoustic field that is reflected by the top wall, which will induce either constructive or destructive interferences.  
\[p\big{(}x_{M}^{-},t\big{)}=2p_{s}(r_{-},\theta_{-},t)+p_{s}\big{(}r_{-}{{}^{ \prime}},\theta_{-}{{}^{\prime}},t\big{)} \tag{37}\]