*4.1. Outdoor Propagation Models*

In this section, we describe the main characteristics of the outdoor propagation model used in this work. We consider a sound signal radiating as a spherical isotropic wave-front [51] through an homogeneous medium; that is, a medium with constant sound speed (*C*0). To define the sound velocity as *c*<sup>0</sup> = 343.23 ≈ 343 m/s, we assume the following conditions:


In such a case, the acoustic free-field intensity of the radiation decreases with the inverse square of the distance and might travel to a receiver along a direct path and a number of reflected paths. Therefore, we can describe the acoustic pressure (*p*(*r*)) at a Euclidean distance *r* from the emitter in the following form [52]:

$$p(r) = \left[\frac{A(r\_0)}{r\_0}\right]e^{jkr\_0} + \sum\_{i=1}^{i=N} Q\left[\frac{A(r\_i)}{r\_i}\right]e^{jkr\_i},\tag{5}$$

where *r*<sup>0</sup> is the distance travelled by the direct path; *ri* the distance travelled by each of the remaining *N* paths, which may be reflected by the ground, surrounding buildings, trees, and so on; *A*(*r*) is the atmospheric attenuation; *Q* is the reflection coefficient; and *k* = *w*/*c*<sup>0</sup> is the wave number.

The atmosphere dissipates sound energy through two major mechanisms—viscous losses and relaxational processes—which have been extensively studied in the ANSI Standard S1-26:1995 [53]. The main mechanism of absorption is proportional to the square of the frequency. The relaxational processes also depend on the relaxation frequency of nitrogen and oxygen. Given the above atmospheric conditions, we can compute the attenuation suffered by an acoustic signal due to atmospheric absorption as:

$$A(r) = \frac{ar}{100} \text{ [dB]},\tag{6}$$

where *r* (in meters) is the distance between emitter and receiver and *α* is the absorption coefficient (in dB/100 m). Given the above meteorological conditions of temperature and relative humidity, we can take the absorption coefficient equal to 0.54 dB/100 m at 1000 Hz and 10.96 dB/100 m at 10,000 Hz.

The reflection coefficient is defined for spherical waves reflecting from complex plane boundary and can be approximated as:

$$\mathcal{Q} = R\_p(\theta) + B(1 - R\_p(\theta))F(w),\tag{7}$$

where *Rp*(*θ*) is the plane-wave reflection coefficient, *θ* is the glancing angle, *B* is a correction term, and *F*(*w*) is the boundary loss function defined by means of the numerical distance *w* [53]. If we assume a locally reacting ground and set *B* = 1, then:

$$R\_p(\theta) = \frac{\sin(\theta) - 1/Z}{\sin(\theta) + 1/Z},\tag{8}$$

where *Z* is the normalised acoustic impedance of the ground. We note that, for low glancing angles (*θ*), *Rp* → −1 and, for high frequencies, *F*(*w*) diminishes and, consequently, *Q* = *Rp*.
