## 3 Parametric Investigation
### Varying the TE Noise Source Location and/or Frequency
#### 3.2.2 Interpretation of the Installation Effects  
The above oscillations of the RMS delta can be explained by the alternance of constructive and destructive interferences between the incident and reflected fields that reach the microphones. Indeed, depending on the TE noise source frequency (i.e., wavelength) and location (i.e., propagation distance), each microphone will see these incident and reflected waves reaching it with a specific phase delay. Whenever the waves are perfectly in phase, their amplitude will sum up and the RMS delta will be higher. Contrarily, if some waves are in phase opposition, they will partly cancel each other, thereby reducing the RMS delta. For the top (resp. bottom) wall microphones, this effect can be inferred using Equation (65) (resp. Equation (66)). In this equation, the two first terms are balanced by a third one, which involves a \(cos(k_{0}\Delta r)\) quantity (with \(\Delta r_{+}=r_{+}-r_{+}\)\" and \(\Delta r_{-}=r_{-}-r_{-}\)\({}^{\prime}\) for the top and bottom walls, respectively). This quantity varies between \(1\) and \(-1\), depending  
Figure 14: Low to high frequency (\(St=[0.294;2.94]\)) noise radiation from the trailing edge of an airfoil set at low (\(\alpha=0^{\circ}\), solid line) or high (\(\alpha=25^{\circ}\), dashed line) incidences. Deltas (in dB) between the incident and total RMS pressure fields, as recorded for two pairs of microphones: M1\({}^{\pm}\) (\(x_{M}/c=0.5\)) and M2\({}^{\pm}\) (\(x_{M}/c=2\)) on top and bottom walls: M1\({}^{+}\) (**a**), M1\({}^{-}\) (**c**), M2\({}^{+}\) (**b**), M2\({}^{-}\) (**d**). on the values taken by the \(2\pi\)-modulo remainder of \(k_{0}\Delta r\). The latter term is driven by the difference in the propagation paths between the incident and reflected waves, which ultimately translates into possible phase delays. Indeed, considering that the wave number is \(k_{0}=\omega/c_{0}=2\pi/\lambda\) with \(\lambda\) the wavelength, one has \(k_{0}\Delta r=2\pi(\Delta r/\lambda)\). The phase delay and subsequent interference effect thus depends on how the difference in propagation paths scales with the wavelength. For instance, whenever the former is an exact multiple of the latter (\(|\Delta r/\lambda|=m\)), then \(cos(k_{0}\Delta r)=1\), i.e., the RMS delta is minimized (given the negative sign of the cosines product, cf. Equations (65) and (66)). Oppositely, when the difference in propagation paths is an odd multiple of half the wavelength (\(|\Delta r/\lambda|=(2m+1)/2\)) then \(cos(k_{0}\Delta r)=-1\), and the RMS delta is maximized (cf. Equations (65) and (66)). As an illustration, let us consider the particular (and simplest) case of M1\({}^{\pm}\) microphones (\(x_{M}=c/2\)) when the airfoil is put at no incidence (\(\alpha=0\)), i.e., the source is located at \(x_{s}=b\), \(y_{s}=0\). Considering that, in the present case \(b=c/2\) as well as \(c=h/2.5\) (cf. Section 2.2.6), the top wall microphone M1\({}^{+}\) is such that (cf. Equations (29), (31) and (65))  
\[r_{+}=\sqrt{\left(x_{M}-x_{s}\right)^{2}+\left(h-y_{s}\right)^{2}}=h \tag{67}\]  
and  
\[r_{+}{}^{\prime\prime}=\sqrt{\left(x_{M}-x_{s}\right)^{2}+\left(y_{s}+3h \right)^{2}}=3h \tag{68}\]  
On the other hand, its bottom wall counterpart M1\({}^{-}\) is such that (cf. Equations (34), (35) and (66))  
\[r_{-}=\sqrt{\left(x_{M}-x_{s}\right)^{2}+\left(h+y_{s}\right)^{2}}=h \tag{69}\]  
and  
\[r_{-}{}^{\prime}=\sqrt{\left(x_{M}-x_{s}\right)^{2}+\left(y_{s}-3h\right)^{2} }=3h \tag{70}\]  
Therefore, for both microphones, the difference in propagation paths becomes \(|\Delta r|=2h\). Consequently, their RMS delta will be minimum for source frequencies such that \(|\Delta r/\lambda|=m\), i.e., \(\lambda=2h/m\), with \(m=1\), \(2\), \(\ldots\), etc. The lowest one corresponds to \(m=1\), i.e., \(\lambda=2h\). Considering the present Strouhal definition (\(St=f\cdot c/c_{0}=c/\lambda\)) and the fact that \(c=h/2.5\), this gives \(St=1/5=0.2\), which matches with the above observations, \(St_{min}=0.2\) mod(\(\Delta S_{t}\)).