## 2 Analytical Model
### Root Means Square (RMS) Pressure
#### 2.3.3 Illustration via a Test Case  
Still regarding the previous test case, Figure 4 depicts the RMS pressure fields recorded within the vein. As previously, each component is isolated on purpose, e.g., the incident field, the reflected fields along with their various interactions, and the resulting total field. Again, the latter differ significantly from the former (compare for instance Figure 4a-d), both quantitatively (higher amplitude levels) and qualitatively (more complex patterns). Figure 4: Low frequency (\(St=0.294\)) noise radiation from the trailing edge of an airfoil set at relatively high incidence (\(\alpha=20^{\circ}\)). RMS pressure fields: (**a**) Incident field (1 st term in Equation (64)); (**b**) top plus bottom wall reflected fields and their interaction (2 nd, 3 rd and last term in Equation (64); (**c**) incident and reflected fields’ interaction (4 th and 5 th terms in Equation (64)); (**d**) total field (\(\widetilde{p}_{RMS}(\mathbf{x})\), all terms in Equation (64)). For visualization purposes, all RMS pressure fields are normalized by \(A_{max}\) and depicted over an amplitude range [0; 2] using 60-iso contours. To assess more precisely their differences, Figure 5 depicts the deltas (in dB) between the incident and total RMS pressure fields. As revealed by this map, the installation effects can be significant depending on the observer location, with deltas ranging from \(-15\) dB (cf. the blue zones) to more than \(30\) dB (cf. the white areas). Notably, the impact is higher in those regions which originally corresponded to the shadow zone of the dipole, namely along its axis (i.e., the projected airfoil centerline). There, the rise of amplitude levels incurred by both the reflection and interference effects leads to deltas of up to \(60\) dB in some points, to begin with those located along the walls. This is more visible in Figure 6, which depicts both the incident and the total RMS pressures recorded along the top and bottom walls (cf. Figures 6a and 6b, respectively), as well as their associated deltas (cf. Figure 6c). Here again, one can see how the incident pressure is modulated by its reflected counterparts and their subsequent interaction, thereby leading to an entirely different total RMS pressure. When considering for instance those points where the incident pressure is maximum (\(x/c\sim 0.8\) and \(x/c\sim 0.0\) of the top and bottom walls, respectively), the total field appears to be roughly \(2.5\) times more important, thereby leading to a delta of approx. \(+8\) dB. Interestingly (even though logically) enough, it is rather in those locations where the incident pressure is close to zero that the delta skyrockets (see for instance what happens at points \(\,|\,x/c\,|\sim 7\) on top and bottom walls, where deltas reach \(+60\) dB and \(+58\) dB, respectively). The above results illustrate well how importantly the reverberation effects can alter the TE noise signature, thereby potentially biasing its in situ measurement using flushed mounted wall microphones. Figure 5: Low frequency (\(St=0.294\)) noise radiation from the trailing edge of an airfoil set at relatively high incidence (\(\alpha=20^{\circ}\)). Deltas (in dB) between the incident and total RMS pressure fields. Figure 6: Low frequency (\(St=0.294\)) noise radiation from the trailing edge of an airfoil set at relatively high incidence (\(\alpha=20^{\circ}\)). Left (**a**) and center (**b**) side, respectively: RMS pressure recorded along the top and bottom walls (depicted in linear scale, once normalized by \(A_{\mathrm{max}}\)). Incident field (long dashes with delta symbols), all reflected fields (small dashes with circle symbols) and total field (solid line with square symbols). Right side (**c**): Deltas (in dB) between the incident and total RMS pressure fields, as recorded along both the top (solid lines) and bottom (dashes) walls.