# A Fast UTD-Based Method for the Analysis of Multiple Acoustic Diffraction over a Series of Obstacles with Arbitrary Modeling, Height and Spacing

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Method

#### 2.1. Selection of Nodes of the Scenario

#### 2.1.1. Case of Knife Edges and Wide Barriers

#### 2.1.2. Case of Polygonal T-Shaped Barriers

#### 2.1.3. Case of Y-Shaped Barriers

#### 2.1.4. Case of Doubly Inclined Barriers

#### 2.2. Selection of Relevant Paths and Obstacles

#### 2.2.1. Funicular Profiles

#### 2.2.2. Relevant Obstacles by Fresnel Ellipsoid Method

_{1}is the distance of any point P from one end of the link and d

_{2}is the distance of P from the other end, as shown in Figure 8. Equation (2) is derived from the application of the Huygens–Fresnel principle to wave propagation analysis [13], and it characterizes the effective ellipsoid shape, with a maximum radius at the midpoint of the path.

_{1i}and d

_{2i}are the distances to the obstacle node located on the left and on the right of the funicular profile, respectively. Following this step, the method assesses whether the frequency is within the frequency range to be analyzed or not.

#### 2.3. Creation of Adjacency Matrix (Graph Theory)

_{i}).

#### 2.4. Pathfinding with the Breadth-First Search (BFS) Method

- first, visit v;
- then, all its adjacent vertices are visited;
- next, those adjacent to the latter are visited, and so on.

#### 2.5. Calculation of Total Sound Pressure Field at the Receiver

^{−4}). Therefore, a threshold level of 0 would imply not discarding any path.

#### 2.5.1. Obstacle Coefficient Factor γ

- $\gamma \left(n\right)=\gamma \left(n-1\right)\xb7{\gamma}_{R}$, ground reflection case, where n is the index of the current obstacle of the path, with$${\gamma}_{R}=\frac{{A}_{R}}{{A}_{D}},{A}_{R}=\frac{{s}_{i}}{{s}_{i+1}+{s}_{i}},{A}_{D}=\sqrt{\frac{{s}_{i}}{{s}_{i+1}\xb7\left({s}_{i}+{s}_{i+1}\right)}}$$
- $\gamma \left(n\right)=\gamma \left(n-1\right)\xb7{\gamma}_{b}$, diffraction over wide barriers case, with [6]$${\gamma}_{b}=\{\begin{array}{c}1,iftheedgesoftheobstacle\\ andthereceiverarenotconnected\\ \\ 0.5,iftheedgesoftheobstacle\\ andthereceiveareconnected\end{array}$$
- $\gamma \left(n\right)=\gamma \left(n-1\right)\xb7{\gamma}_{c}$, diffraction over cylinders case, with$${\gamma}_{c}=\{\begin{array}{c}1,\hspace{1em}LOSbetweenTxandRx,litregion\hfill \\ \\ \frac{{A}_{D1}}{{A}_{D2}}\xb7\frac{{e}^{-ik\left(ta+s{s}_{i+1}-{s}_{i+1}\right)}}{{e}^{-ik\left({s}_{i}-s{s}_{i}\right)}},\hspace{1em}shadowregion\hfill \end{array}$$

- ss
_{i}is the distance of the incident ray impinging on the cylinder; - ss
_{i+1}is the distance of the diffracted ray from the cylinder to the next end; - and ta is the arc length traveled by the creeping wave over the cylinder from the incidence point to the exit one.

#### 2.5.2. Diffraction Coefficient for Knife Edges, Wedges and Wide Barriers

#### 2.5.3. Diffraction Coefficient for Cylinders (Rounded Obstacles)

_{i}and s

_{j}are the slant ranges, as shown in Figure 13.

_{2}(t) are defined as:

_{i}(t) as a Miller-type Airy function:

#### 2.5.4. Ground Reflection Coefficient

## 3. Results and Discussion

#### Analysis of Computational Efficiency of the Proposed Method

^{−5}). It should be noted that three wide barriers imply a maximum number of 12 nodes and 11 hops, including reflections.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Example of extraction of nodes from a scenario built by three knife edges, one T-shaped obstacle, one wide barrier, two wedges and one cylinder. The interior angles of the wedges are 20° (wedge on the left) and 10° (wedge on the right). The radius of the cylinder is 0.2 m. The source and the receiver appear in red in the profile.

**Figure 7.**Funicular profile created for the extracted scenario of the example in Figure 2.

**Figure 8.**Consideration of obstacles 3, 4 and 7 within the filtered profile by applying the Fresnel zone criteria (with n = 1 and a minimum frequency of 100 Hz).

**Figure 10.**Selection of obstacles and reflection points (nodes) in the profile of Figure 2 at 100 Hz. At frequencies higher than 123.6 Hz, the obstacle renamed as node 6, as well as the reflection node 7 will also be ignored.

**Figure 11.**General line of sight (LOS) path directed graph (digraph). Funicular nodes (F

_{i}), in red, including source and receiver, are common nodes for any path.

**Figure 15.**Graphical user interface of the software tool. Scenario of Figure 2 created on the right display.

**Figure 16.**IL (dB) of the scenario of the example of Figure 2, with (red line) and without (black line) reflections on the ground.

**Figure 20.**IL map for the same scenario as in Figure 17a for 1 kHz (double knife edge barrier). The colored scale on the right is in dB.

**Figure 21.**IL at the receiver for the scenario of Figure 17c with two wide barriers. The numerical results obtained with the BEM are also shown in the plot.

**Figure 22.**IL at the receiver for the scenario of Figure 17d with three wide barriers. The numerical results obtained with the BEM are also shown in the plot.

**Figure 23.**IL at the receiver for the scenario of Figure 17e with three wide barriers. The numerical results obtained with the BEM are also shown in the plot.

**Figure 24.**IL at the receiver for the scenario of Figure 17f with a Y-shaped barrier. The numerical results obtained with the BEM are also shown in the plot.

**Figure 25.**IL at the receiver for the scenario of Figure 17g with a T-shaped barrier. The numerical results obtained with the BEM are also shown in the plot.

**Figure 26.**IL at the receiver for the scenario of Figure 17h with a doubly inclined barrier. The numerical results obtained with the BEM are also shown in the plot.

**Figure 27.**Comparison of measurements, proposed method simulations and results obtained by Berry for a cylinder of a radius of 5 m, as shown in Figure 17i.

**Figure 28.**Comparison of measurements, proposed method simulations and results obtained by Berry for a cylinder of a radius of 5 m, as shown in Figure 17j.

**Figure 29.**Scenario with ten cylinders randomly located between the tx and rx. The Fresnel ellipsoids at 100 Hz are superimposed in the profile.

**Figure 30.**Comparison of the scenario depicted on top assuming knife edges and cylinders with decreasing radii. Reflections on the ground are not considered in this case.

**Figure 31.**Comparison of IL for scenario of Figure 17e with Min and Qiu’s prediction, BEM and the proposed method when decreasing the number of possible hops in the path.

**Figure 32.**Complex scenario with nine different barriers. One is discarded by Fresnel criterion at the distance of 15.2 m.

**Figure 33.**IL spectrum for scenario of Figure 32 with and without reflections.

**Table 1.**Parameters for Scenario of Figure 32.

Parameter | Result |
---|---|

Subbands [Hz]—by Fresnel ellipsoid analysis | [200–565.4], [565.4–646.9], [646.9–684.3], [684.3–1293.8], [1293.8–5000] |

Final selection of nodes | 20 (max. at lowest band) down to 12 (at highest band) |

Hops selected | 19 (max at lowest band) down to 11 (at highest band) |

Paths selected | 1800 (max. at lowest band) down to 20 paths (at highest band) |

Frequency band [Hz] | 200–5000 |

Frequency bins | 1500 |

Minimum relative threshold level | 10-5 |

Elapsed time without reflections [s] | 1.81 |

Elapsed time with reflections [s] | 19.02 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Pardo-Quiles, D.; Rodríguez, J.-V. A Fast UTD-Based Method for the Analysis of Multiple Acoustic Diffraction over a Series of Obstacles with Arbitrary Modeling, Height and Spacing. *Symmetry* **2020**, *12*, 654.
https://doi.org/10.3390/sym12040654

**AMA Style**

Pardo-Quiles D, Rodríguez J-V. A Fast UTD-Based Method for the Analysis of Multiple Acoustic Diffraction over a Series of Obstacles with Arbitrary Modeling, Height and Spacing. *Symmetry*. 2020; 12(4):654.
https://doi.org/10.3390/sym12040654

**Chicago/Turabian Style**

Pardo-Quiles, Domingo, and José-Víctor Rodríguez. 2020. "A Fast UTD-Based Method for the Analysis of Multiple Acoustic Diffraction over a Series of Obstacles with Arbitrary Modeling, Height and Spacing" *Symmetry* 12, no. 4: 654.
https://doi.org/10.3390/sym12040654