## 2 Jet Noise Prediction Model
### Distribution Source Method  
The Distribution Source Method (DSM) mentioned in [PERSON]'s work [11] assumes that the jet noise is modeled by allocating discrete noise sources along the exhaust stream. Since the sound pressure level (SPL) spectrum varies in frequency along the length of the plume as a function of jet characteristics, rocket engine geometry, and position on the launch pad, each discrete noise source will have to act on a specific frequency band with a unique sound power level depending on its location. Figure 2 schematically represents the rocket engine, the set of acoustic sources distributed along the exhaust stream (distance \(l_{b}\) from the nozzle exit measured along the \(x\)-axis), and the relative position of the observer (distance \(r\) from each source). The angle \(\theta\) is defined for each source by the line between the source and the observer (point \(P\)) and the jet axis. Finally, the distance \(s\) and the angle \(\phi\) will be useful to define a unique reference system centered at the nozzle exit. According to [PERSON] [11], the overall acoustic power \(W_{OA}\) in watts is estimated using:  
\[W_{OA}=\eta\frac{nFUL_{e}}{2} \tag{1}\]  
where \(\eta\) is the ratio between the sound power and the exhaust's mechanical power, termed acoustical efficiency, \(n\) is the number of nozzles, \(F\) is the thrust of each engine in Newtons, and \(U_{e}\) is the fully expanded jet velocity in meters per second. For highly supersonic jets the sound power is proportional to the mechanical power generated by the engine (proportional to the product of the thrust and the jet velocity), thus, acoustical efficiency result in a constant value. For undeflected standard chemical rocket exhaust, the latter falls in the range of 0.2% and 1%. The overall sound power level \(L_{W}\), which specifies the power delivered in decibels relative to one picowatt, is given by:  
\[L_{W}=10 log_{10}\left(\frac{W_{OA}}{10^{-12}}\right) \tag{2}\]  
Each rocket engine produces a unique sound power spectrum, depending on the physical and chemical properties of the exhaust plume, the geometric characteristics of the nozzle, the thrust, etc. However, a dimensionless analysis allows to obtain a single acoustic spectrum for a wide range of engines, more precisely, the normalized power spectra are weakly dependent on their parameters, with the exception of the Strouhal number. The latter is a dimensionless number used to describe oscillating flow mechanisms, defined as the product of the frequency and the nozzle exit diameter divided by the fully expanded exit velocity. Figure 2: Schematic representation of the noise sources distribution according to [PERSON]’s method. Figure 3a illustrates how this result has been confirmed by a large number of experiments, while Figure 3b shows the interpolated data for the case study (low-thrust HRE):  
The normalized spectrum is then converted to a conventional acoustic power level defined in a generic bandwidth \(\Delta f_{b}\) by the following relationship [11]:  
\[L_{W,b}=10 log_{10}\left[\frac{W(f)}{W_{OA}}\frac{U_{e}}{d_{e}}\right]+L_{W}-10 log_{10}\left(\frac{U_{e}}{d_{e}}\right)+10 log_{10}(\Delta f_{b}) \tag{3}\]  
where \(L_{W,b}\) represents the sound power level in dB in the band centered on frequency \(b\), while \(d_{e}\) is the exit nozzle diameter in meters. A single discrete noise source is assigned a band sound power level, whose strength is defined by Equation (3), and then arranged along the exhaust flow. How these are moved along the plume axis depends on the similarity curves in Figure 4a. In particular, the Strouhal numbers calculated above are useful to interpolate the reference curve for the single-nozzle (undeflected) chemical rocket and estimate the apparent source positions (Figure 4b).