## 2 Model Development
### Propagation Filtering  
The sound propagation model described in this section incorporates the following effects:  
* Propagation delay
* Doppler effect (frequency shift and amplification)
* Convective amplification
* Geometrical spreading
* Ground reflection
* Air absorption  
Other outdoor sound propagation effects that may be relevant in certain situations are screening [37; 38; 39], foliage attenuation [38], meteorological effects due to an inhomogeneous atmosphere [38; 39; 40; 41; 42], as well as reflections at artificial [38; 43] and natural surfaces [44; 45; 42]. Most published environmental noise auralization models simulate some of the above listed effects by applying a 1/3 octave filter bank and adjusting the filter gains [8; 14; 15]. In this model, however, all of these effects are applied in the time domain, _i.e._, by time-variant digital filters. Sound propagation is modeled by two paths, namely for direct sound and a single ground reflection (in the following account indicated by subscripts \"dir\" and \"gr\", respectively). The sound pressure of a point source has a \(1/r\) distance dependency. Thus, to model geometrical spreading, the emitted sound pressure signals, \(x\), are divided by their path length \(r_{\text{dir}}\) or \(r_{\text{gr}}\), respectively. The interaction of a sound wave with the ground influences its amplitude and phase as a function of frequency. This effect can be modeled by convolution of the ground-reflected signal with a time-variant filter [46]. Furthermore, the attenuation due to air absorption can be efficiently modeled using a filter [46]. Considering these aspects, the receiver signal \(y\) is calculated by:  
\[y(t^{\prime})=h_{\text{air},t^{\prime}}(t^{\prime})*\left(\frac{x_{\text{dir}} (t^{\prime})}{r_{\text{dir}}(t^{\prime})}+h_{\text{gr},t^{\prime}}(t^{\prime})* \frac{x_{\text{gr}}(t^{\prime})}{r_{\text{gr}}(t^{\prime})}\right) \tag{20}\]  
where \(*\) denotes linear convolution, \(t^{\prime}\) the receiver time axis, \(r_{\text{dir}}\) is the source-receiver distance, \(r_{\text{gr}}\) is the distance source-ground reflection point-receiver, \(x_{\text{dir}}\) and \(x_{\text{gr}}\) are delayed versions of the emitted sounds and \(h_{\text{air},t^{\prime}}\) and \(h_{\text{gr},t^{\prime}}\) denote impulse responses of time-dependent filters described in Sections 2.3.2 and 2.3.3. The modeling of effects due to source motion and the propagation delay are explained in Section 2.3.1. Note that the immission angle \(\theta(t^{\prime})\), which is needed for surround reproduction, has to be evaluated on the receiver axis, as well.  
Figure 3 shows normalized spectrograms of two synthesized pass-by sounds. For the synthesis of the sound of propulsion, the respective engine condition courses depicted in a and b of Figure 3 were used. They are described in Section 2.2.2. The temporal behavior of the engine states, the spectrograms, as well as the auralizations are published as supplementary data (see the videos in the Supplementary File). In both simulations, the car passes the receiver at Time 0 at 30 km/h. The receiver is located 1.2 m above a hard ground at a distance \(D=7.5\) m. As at the pass-by, the engine speed still increases, the Doppler frequency shift is not directly observable in the course of the order frequencies. However, the gear change moments can be well observed as local decreases of order frequencies. As a consequence of the used engine condition courses, these frequency drops occur at higher frequencies for the sporty driving style (d).