## Materials and Methods
### Theoretical Method
#### 2.3.1 Pressure Diffraction Coefficients (Dn)  
According to UTD [45], the pressure diffraction coefficient for knife-edges and wedges can be defined as:  
\[D(v,k,L,s1,s2,\theta_{2},\theta_{1})=\frac{-e^{-i\frac{\pi}{4}}}{2v\sqrt{2\pi k }}\left[ \begin{array}{c}tan^{-1}\bigg{(}\frac{\pi+(\theta_{2}-\theta_{1})}{2v }\bigg{)}F(kLa^{+}(\theta_{2}-\theta_{1},v))\\ +tan^{-1}\bigg{(}\frac{\pi-(\theta_{2}-\theta_{1})}{2v}\bigg{)}F(kLa^{-}( \theta_{2}-\theta_{1},v))\\ +R_{n}\cdot tan^{-1}\bigg{(}\frac{\pi+(\theta_{2}+\theta_{1})}{2v}\bigg{)}F( kLa^{+}(\theta_{2}+\theta_{1},v))\\ +R_{0}\cdot tan^{-1}\bigg{(}\frac{\pi-(\theta_{2}+\theta_{1})}{2v}\bigg{)}F( kLa^{-}(\theta_{2}+\theta_{1},v))\end{array} \right] \tag{3}\]  
where \(R_{0}\) and \(R_{n}\) represent the reflecting coefficients for adjacent (0) and opposite (\(n\)) obstacle faces seen by the incident wave. The reflecting coefficients \(R_{0}\) and \(R_{n}\) will depend on: the incident (\(\theta_{1}\)) and diffracting (\(\theta_{2}\)) angles of the path between the source and receiver, respectively; and the normal specific acoustic impedances of the faces seen by them (Figure 9). Further explanations are provided in Section 2.3.2, and [45] offers greater detail regarding all these parameters:  
Figure 8: PARDOS (Péridias Acüsticas por Reflexión y Difraccón de la Onda Sonora) main GUI (graphical user interface) with ‘About’ pop-up window. F[x] represents the \"transition function\", which can be defined as a Fresnel integral [45]:  
\[F[x]=2i\sqrt{x}e^{ix}\int_{\sqrt{x}}^{\infty}e^{-i\hbar^{2}}du, \tag{4}\]  
\[L=\frac{s_{i}s_{j}}{s_{i}+s_{j}} \tag{5}\]  
and  
\[a^{\pm}(\beta)=2\cos^{2}\!\left(\frac{2v\pi N^{\pm}-\beta}{2}\right),\ \beta= \theta_{2}\pm\theta_{1} \tag{6}\]  
where \(N^{\pm}\) are the integers that most nearly satisfy the equations  
\[2\pi vN^{+}-\beta=\pi, \tag{7}\]  
\[2\pi vN^{-}-\beta=-\pi. \tag{8}\]  
In the same way, we may employ UTD to explain the diffraction coefficient with cylindrical structures using a pair of scattering mechanisms, either being the field's diffraction or reflection components [49; 50]. Thus with the \"shadow region\" (source-receiver line-of-sight (LoS) absent), we may consider this diffraction coefficient:  
\[T_{s,h}(a)=-m_{p}\sqrt{\frac{2}{k}}e^{-i\frac{\pi}{4}}e^{-i\pi(a)}\left\{ \frac{-F[X(a)]}{2\varepsilon(a)\sqrt{\pi}}+[q^{*}(\varepsilon(a))]\right\}, \tag{9}\]  
being  
\[a=\pi+\alpha+\beta,\ \ \alpha,\beta\geq 0 \tag{10}\]  
\[X(a)=\frac{kL(a-\pi)^{2}}{2}, \tag{11}\]  
\[\varepsilon(a)=m_{p}(a-\pi), \tag{12}\]  
\[F[x]=2i\sqrt{x}e^{ix}\int_{\sqrt{x}}^{\infty}e^{-i\hbar^{2}}du, \tag{4}\]0  
\[F[x]=2i\sqrt{x}e^{ix}\int_{\sqrt{x}}^{\infty}e^{-i\hbar^{2}}du, \tag{4}\]1  
where \(\alpha\) and \(\beta\) are the angles of the arc run by the 'creeping' wave over the rounded surface; \(k\) is the wavenumber; \(r_{obs}\) is the radii of the cylinder; and \(\mathbf{s_{i}}\) and \(\mathbf{s_{j}}\) are the slant ranges from the source and receiver, respectively. \[F[x]=2i\sqrt{x}e^{ix}\int_{\sqrt{x}}^{\infty}e^{-i\hbar^{2}}du, \tag{4}\]2 Notation used in a single diffracting wedge. Reflection coefficients for the 0 face, with incidence angle \(\theta_{1}\), and for the \(n\) face, with reflection angle \(v\pi\!-\!\theta_{2}\). A comprehensive description of each variable is explained in [49; 50] as well as in [43]. The first addend of Equation (9) describes the Fresnel diffraction process. _F[x]_ represents the \"transition function\" already defined in Equation (4). For the second extension of Equation (9), the term \(q^{\star}(\varepsilon(a))\) represents the \"Fock scattering function\", which deals with 'creeping' waves generated along a smooth body's surface (e.g., spheres or cylinders [49]).