## 2 Jet-Noise Source Model and Propagation  
The complexity of the detailed modeling of the acoustic component in rocket exhaust can be circumvented using empirical approaches, on the basis of the assumption that an equivalent source distribution can be determined using few parameters (e.g., jet exit conditions and acoustic efficiency) and some universal curves determined experimentally. In particular, the first Eldred model [8] is based on the observation that each slice of the jet contributes to acoustic radiation, mostly in the frequency band \(f_{i}\) that depends on the distance of the slice from the exhaust. The subsequent reformulation of the approach states that each slice contributes to the full noise spectral frequency content with an overall power that depends on its curvilinear distance from the exhaust [12]. Differently from the original Eldred model, which directly provides a definition of the sound pressure level (SPL), its reformulation can be revised to provide an explicit formulation of the acoustic pressure phasor \(p_{j}\)[14]. In particular, this simpler version of the reformulated semiempirical model moves from the relation  
\[p_{j0}=g(r_{j})\sqrt{P_{0}\cdot 8\pi\rho c} \tag{1}\]  
equating the phasor of the incident pressure, \(p_{j0}\), due to the \(j\)th source, to the corresponding monopole amplitude, \(Q_{0}=\sqrt{P_{0}\cdot 8\pi\rho c}\), with \(P_{0}\) indicating the power, \(\rho\) being the air density and \(c\) being the speed of sound at ambient conditions, through the green function of the nonconvective Helmholtz operator, \(g(r_{j})\); the directivity effect produced by the inflow refraction, can be taken into account with a directivity function \(\mathrm{D}=\frac{I}{I_{0}}\), where \(I_{0}=\frac{P_{0}}{4\pi r^{2}}\) and \(I\) indicates the intensity. Hence, the pressure of each monopole source, composing the equivalent set that reproduces the jet noise, can be expressed in the form  
\[p_{j}=g_{j}(r)Q_{0}\sqrt{\mathrm{D}_{\mathrm{j}}(\theta)} \tag{2}\]  
with \(\theta\) indicating the angle between the observer and the monopole source position. The second formulation of the Eldred model assumes a contribution from each jet slice to the full noise spectral frequency content with an overall power that depends on its curvilinear distance from the exhaust [12]. Denoting \(w(s_{k})\) as the sound power per unit axial length, the contribution of each slide reads as  
\[w(s_{k})\Delta s_{k}=\frac{L_{ref}w(s_{k})}{W_{ac}}W_{ac}\frac{\Delta s_{k}}{L _{ref}} \tag{3}\]  
where \(L_{ref}\) is a jet reference length, and \(k\) denotes the \(k\)th slice at a distance \(s_{k}\) from the exhaust. \(W_{ac}\) represents the overall acoustic power [15] of the engine, which is a fraction of the mechanical power \(W_{m}\) (\(W_{ac}\simeq\ \eta W_{m}\) with \(\eta=0.03\)). The overall power density level, \(w(s_{k})\), can be distributed on the frequency spectrum. Denoting with \(w(s_{k},f_{i})\) the sound power level per unit axial length and unit frequency, one can introduce the reduced sound power density:  
\[\tilde{w}=\frac{w(s_{k},f_{i})}{w(s_{k})}\frac{u_{j}}{s_{k}}\frac{c_{a}}{c_{j}} \tag{4}\]  
with \(u_{j}\) being the jet speed at the exhaust, \(c_{a}\) being the ambient sound speed and \(c_{j}\) being the jet exhaust speed. Consequently, it is possible to obtain the monopole source's acoustic power at each axial length and for each frequency from  
\[P_{ik}=\tilde{w}\cdot\tilde{w}\cdot W_{ac}\cdot\frac{s_{k}}{u_{j}}\frac{c_{j} }{c_{a}}\frac{\Delta s_{k}\Delta f_{i}}{L_{ref}} \tag{5}\]  
The revised Eldred-based model is enriched with two free parameters, \(\alpha\) and \(\beta\), respectively related to the mixing-layer turbulence length scale and its integral correlation length. The Eldred standard model describes the jet noise with a set of equivalent sources, laying along the jet axis with no consideration of their number or spatial distribution. It is reasonable to think that the spatial scale of each source, and consequently its number, is related to its turbulent length scale.