## 2 Analytical Model
### Instantaneous Pressure
#### 2.2.6 Illustration via a Test Case  
To illustrate the above developments, let us consider a 2D test case that is representative of a typical airfoil TE noise experiment that would be conducted within a closed-vein facility of half-height \(h\) (cf. Figure 1). The airfoil has a chord \(c=h/2.5\) and it is rotated around its mid-chord point (\(b=c/2\)) with an angle of attack \(\alpha=20^{\circ}\). Let us assume that, under those conditions, the airfoil emits a strong, tonal, dipolar TE noise emission of frequency corresponding to a Strouhal number \(St=f\cdot c/c_{0}=0.294\) (with \(c_{0}\) the sound speed, \(c_{0}=340\) m/s). In real life, this test case could correspond to a small-scale experiment for which a 10 cm chord airfoil would be blown within a vein of height \(2h=0.5\) m, thereby radiating a TE self-noise of frequency \(f=1\) kHz (which is consistent with typical observations from the literature). Notably, since the wavenumber can be expressed as a function of the Strouhal (\(k_{0}=2\pi\cdot St/c\)), the far-field condition invoked in the dipole approximation (\(k_{0}\cdot r>1\), cf. Section 2.2.5) translates into \(2\pi\cdot St\cdot(r/c)>1\). In the present case (\(St=0.294\)), this condition is verified for all observers such that \(r/c>0.54\). Figure 2 depicts the instantaneous pressure field corresponding to an emission time of 10 periods (\(t=10\)T), with such field being decomposed into several components, for illustration purposes. First, the incident field clearly reveals how the TE dipolar source emits following a very directive fashion, with radiation lobes that are oriented orthogonally to the \(\alpha\)-inclined airfoil center line (see Figure 2a). This incident field is then reflected by  
the top wall, leading to backscatter patterns which resemble to the radiation of a virtual source that would mirror the real one (see Figure 2b). Such reflected field finally sums up with its bottom wall counterpart, thereby leading to the strong interference patterns that translate the vein's installation effects (see Figure 2c). Once cumulated, these various components lead to the total pressure field (see Figure 2d), which differs significantly from the original one (compare for instance Figure 2d to Figure 2a). Notably, this resulting overall pressure field reveals specific features that are characteristic of the physics of acoustic propagation in ducts. For instance, one can clearly distinguish various sets of standing waves that are distributed along the \(y\)-axis, whereas being more prominent nearby the source region (\(x/h\sim 0.2\)). From a physical viewpoint, these \(y\)-aligned standing waves obviously originate from the top and bottom walls' cumulative reflections. From a mathematical perspective, they stem from the specific modal basis that is typical of acoustic propagation in 2D ducts. The latter basis is indeed characterized by an infinite set of modes (\(n\): \(0\rightarrow\infty\)), each one being allotted a specific \(y\)-standing wave component (\(\sim\!cos\{n\pi y/(2h)\}\)) and \(x\)-propagative feature (\(\sim\!e^{i(\omega t-k_{x}x)}\)). Notably, the \(x\)-propagative feature is driven by the axial wavenumber, which may be either real (propagative mode) or imaginary (evanescent mode). Whereas all modes are present nearby the source, only the propagative ones can subsist further away from it--which explains the progressive loss of acoustic levels that one can here observe upstream and downstream from the airfoil TE. All this illustrates well how the reverberation effects incurred by the vein lead to a complete redistribution of the noise radiation patterns. This is made more explicit in Figure 3, which depicts the incident, the overall reflected and the total instantaneous pressure fields recorded along either the top or the bottom walls. As can be seen, the interplay between the incident and reflected fields is more or less important, depending on the location.