## 2 Analytical Model
### Instantaneous Pressure
#### 2.2.5 Particular Case of a Harmonic Dipolar Source  
Let us consider the particular albeit common situation where the TE noise source corresponds to a harmonic dipole pulsating with a frequency \(f\). It emits acoustic waves that then propagate within the (supposedly quiescent) medium, being characterized by a wavenumber \(k_{0}=\omega/c_{0}\) where \(\omega=2\pi f\) is the source pulsation. Under a far-field assumption (i.e., \(k_{0}r>1\)), the resulting incident field can be approximated as follows:  
\[p_{s}(r,\theta,t)\approx Re\Big{\{}A(r)B(\theta)e^{i\left(k_{0}r-\omega t \right)}\Big{\}} \tag{38}\]  
In the above expression, \(A(r)\) stands for the amplitude whereas \(B(\theta)\) relates to the directivity, with \(\theta\) the angle subtended by the observer to the dipole axis (cf. Figure 1). For a dipole under a far-field condition (compact source), the \(B(\theta)\) directivity is that of a cardioid, i.e., the acoustic radiation is maximum along the dipole axis, and nil in the orthogonal direction:  
\[B(\theta)=cos(\theta) \tag{39}\]  
When the dipole corresponds to a point source of strength Q, it radiates in a 3D fashion, thereby emitting waves of maximum amplitude:  
\[A^{3D}(r)=i\Big{(}\frac{\omega\rho_{0}}{4\pi r}\Big{)}Q \tag{40}\]  
If, however, the dipole rather corresponds to a line of sources, it radiates in a 2D fashion, and the amplitude becomes:  
\[A^{2D}(r)=i\Big{(}\frac{1}{2}\sqrt{\frac{\omega\rho_{0}}{2\pi r}}e^{i\left( \frac{\pi}{4}\right)}\Big{)}Q \tag{41}\]  
This last case corresponds to the present problem, for which the airfoil trailing edge noise acts as a 2D dipole located at the TE. One can thus express the source field \(p_{s}\) as follows:  
\[p_{s}(r,\theta,t)\approx A_{max}\Big{[}\frac{cos(\theta)}{\sqrt{r}}\Big{]} Re\Big{\{}ie^{i\left(k_{0}r-\omega t+\frac{\pi}{4}\right)}\Big{\}} \tag{42}\]  
Or, when equivalently expressed in non-complex quantities,  
\[p_{s}(r,\theta,t)\approx A_{max}\Big{[}\frac{cos(\theta)}{\sqrt{r}}\Big{]}sin \Big{(}k_{0}r-\omega t+\frac{\pi}{4}\Big{)} \tag{43}\]In the above, \(A_{max}\) refers to the maximum amplitude, which is driven by the frequency (on which depends the source 'compactness', through the wavelength):  
\[A_{max}=\left(\frac{1}{2}\sqrt{\frac{\omega\rho_{0}}{2\pi}}\right) \tag{44}\]  
Reflected and Total Fields  
In this particular case of a harmonic dipolar source, the reflected field becomes  
\[p_{ref}(\textbf{x},t)=p^{\prime}_{ref}(\textbf{x},t)+p^{{}^{\prime\prime}}_{ ref}(\textbf{x},t) \tag{45}\]  
with  
\[p^{\prime}_{ref}(\textbf{x},t)=p_{s}\big{(}r^{\prime},\theta^{\prime},t\big{)} \approx A_{max}\bigg{[}\frac{cos(\theta^{\prime})}{\sqrt{r^{\prime}}}\bigg{]} sin\Big{(}k_{0}r^{\prime}-\omega t+\frac{\pi}{4}\Big{)} \tag{46}\]  
and  
\[p^{{}^{\prime\prime}}_{ref}(\textbf{x},t)=p_{s}(r^{\prime\prime},\theta^{ \prime\prime},t)\approx A_{max}\bigg{[}\frac{cos(\theta^{\prime\prime})}{ \sqrt{r^{\prime\prime}}}\bigg{]}sin\Big{(}k_{0}r^{\prime\prime}-\omega t+ \frac{\pi}{4}\Big{)} \tag{47}\]  
The above reflected fields (Equations (46) and (47)) will sum up with the incident one (Equation (43)), thereby leading to the total pressure field. Incident, Reflected and Total Pressure Fields at the Wall  
For an observer located on the top wall, the resulting overall pressure is obtained by replacing the expression of Equation (32) with those appearing in Equations (43) and (47). \[B(\theta)=cos(\theta) \tag{39}\]0  
Similarly, for an observer located on the bottom wall, the resulting overall pressure is obtained from Equation (37) using expressions from Equations (43) and (46). \[B(\theta)=cos(\theta) \tag{39}\]1