## 3 Description of SST and Its Departure Noise Modeling
Therefore, no attempt has been made to account for the effects of airframe noise during the certification departure procedure of the SST in this study. On the other hand, the effects of noise propagation from the aircraft to the certification measurement points have to be rigorously simulated, because the implementation of VNRS for SST noise reduction to a large extent relies on and benefits from these effects, especially the effect of lateral attenuation discussed in Section 2. Therefore, noise propagation modeling includes spherical spreading, Doppler shift and convective amplification, atmospheric absorption (ISO 9613-1:1993) [33], lateral ground attenuation [34], and ground reflection [35]. It can be seen that the effects of ground reflection and lateral attenuation are calculated independently, which is a standard practice. Recently, a novel method was proposed in [36] to account for both these effects simultaneously. In appendices of [36], detailed descriptions of a ground reflection method and a lateral attenuation method are conveniently provided. However, the authors of the present study do not have previous experience in the implementation of the novel method, nor have they validated it in the experiments; therefore, it has been decided to perform the present study with the use of the standard calculation methods for ground reflection and lateral attenuation. The resulting approach for noise calculation is very robust and is based on a reliable set of data, as well as on the validated and widely used methods for prediction of noise propagation from the source position to the measurement points. Validation of the approach to model SST noise was performed in the context of modeling the noise of a Supersonic Technology Concept Airplane (STCA) developed by NASA [37] (p. 40). TsAGI used SOPRANO software v.4.0 [38] to calculate STCA noise for the reference departure procedure, and its results generally agreed with those provided by NASA and JAXA [39] (Table 2). The flight trajectory of the SST at takeoff is calculated based on a set of two-dimensional flight dynamics equations that describe the horizontal and vertical motion (\(X,H\)) of the \begin{table}
\begin{tabular}{l c c c} \hline  & **NASA** & **TsAGI** & **JAXA** \\ \hline Flyover & 84.4 & 84.3 & 83.5 \\ Lateral (K\({}_{2}\)) & 97.2 & 95.4 & 96.7 \\ Approach & 95.9 & 98.8 & 97.0 \\ Cumulative & 277.5 & 278.5 & 277.2 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of calculation results for STCA certification noise (in EPNdB). aircraft as a function of time \(t\). The rate equations for the position vector \((X,H)\), velocity \(V\), and the climb angle \(\gamma\) are given by  
\[ \begin{array}{c}\frac{dV}{dt}\ =\ \frac{\delta}{W}\big{[}T\cos(\alpha\ +\ \delta)-D-F_{ fric}-W\sin\gamma\big{]},\\ \frac{d\gamma}{dt}\ =\ \frac{\delta}{V\eta}[T\sin(\alpha\ +\ \delta)+L\ -\ W\cos\gamma],\\ \frac{dH}{dt}\ =\ V\sin\gamma,\ \ \frac{dX}{dt}\ =\ V\cos\gamma\end{array}  \tag{2}\]  
where \(g\) is the acceleration due to gravity, \(T\) is the full thrust of the engines, \(W\) is the weight of the aircraft, \(D\) is the aerodynamic drag of the aircraft, \(L\) is the lift force of the aircraft, \(F_{ fric}\) is the rolling friction force, \(H\) is the flight altitude, \(X\) is the distance from the brake release, \(\alpha\) is the angle of attack, \(\delta\) is the angle of the thrust vector direction. Lift and drag forces depend on velocity \(V\) and angle of attack \(\alpha\):  
\[L\ =C_{L}\big{(}\alpha,\theta_{haps}\big{)}\frac{1}{2}\rho V^{2},\ \ \ D\ =C_{D}\big{(}\alpha,\theta_{haps}\big{)}\frac{1}{2}\rho V^{2} \tag{3}\]  
where \(\rho(H)\) is the air density, while coefficients \(C_{L}\) and \(C_{D}\) are considered known for a given aircraft configuration defined by the position of high lift devices \(\theta_{haps}\)[40].