## 3 Propeller Aeroacoustics: Theoretical Background  
The design of an optimal propeller involves a large number of design variables to maximise its efficiency. When the design of a propeller array is considered, those variables become an even larger number. Despite the differences on the final design (i.e., blade geometry, number of blades, number of propellers) and its applications, the contributions to the noise-generating process are independent of the configuration type [33]. The acoustic signature of a propeller is the combination of a harmonic spectrum and a broadband one. The harmonic noise, also called tonal noise, is the periodic component which repeats itself exactly at each propeller revolution. For an observer not rotating with the blades, the period of the tonal noise, and thus its frequency is linked to the number of blades (\(n_{b}\)) and the rotational speed (\(\Omega\)) as follows  
\[T=\frac{1}{\Omega n_{b}}\implies f=\Omega n_{b}, \tag{1}\]  
where \(f\), called _Blade-Passage Frequency_ (BPF), is the fundamental frequency of the tonal noise. Contrarily, broadband noise has a random characteristic containing components at all frequencies with a shape that depends on the propeller installation and operating condition at the most. However, the broadband amplitude on the spectrum is much lower than the tone signal. As depicted in Figure 2, the noise mechanisms that contribute to propeller noise spectrum can be classified in three main groups: steady sources, unsteady sources, and random sources. \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Value** \\ \hline
**No. of blades, \(n_{b}\)** & 16, evenly spaced \\
**Hub radius, \(r_{0}^{H}\)** & 0.124 m \\
**Chord, \(c\)** & 0.297 m \\
**Blade root, \(r_{0}\)** & 0.123 m \\
**Diameter, \(d\)** & 1.484 m \\
**Blade taper ratio, \(\lambda\)** & 1.0 \\
**Pitch angle, \(\hat{\theta}\)** [deg]** & \(90-(-7.6r^{3}-32.47r^{2}+94.45r-10.56)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristic data of propellers. Figure 2: Noise generation contributions to typical propeller spectrum. The steady sources are constant in time in the rotating blade system (i.e., a system fixed with the rotating blade), and therefore they are inherently periodic in the non-rotating one. Sources of this type are linear thickness noise, steady loading noise, and the non-linear quadrupoles noise. The thickness noise is associated with the periodic displacement of the air volume due to a passing blade. It is one of the major noise sources in propeller applications, and it can be represented by a monopole source distribution over the blade surface. The magnitude of this noise component is proportional to the blade volume, and its magnitude increases proportionally to the flight speed. The spectrum of the thickness noise strongly depends on the rotor angular velocity and the shape of the blade cross-section. The linear loading is due to the pressure field acting on the blades during the propeller motion. This noise source is dominant at low to moderate flight speeds. It is generally modelled by dipoles distribution to represent the pressure disturbance that propagates as noise in the medium, with harmonic content at frequencies multiple of the BPF. Finally, the quadrupoles are mainly used to account for the nonlinear effects that may be significant when the blades are in transonic condition and all the viscous and propagation effects not included in the previously described noise sources. Thus, the presence of volume quadrupoles enhances the noise signature obtained by using loading and thickness like sources. Non axial-flow conditions (e.g., due to non-axial flight, installation effects, aerodynamic interaction between adjacent propellers, etc.) result in unsteady blade loading which causes another important noise contribution, named the unsteady loading source.