## Materials and Methods
### Propagation Environment  
The four propagation schemes considered in this work are illustrated in Figures 3 and 4. These schemes consist of:  
* Noise sources (Tx1 and Tx2, respectively) that radiate sound spherically at a height (h\({}_{\text{tx1}}\)) of 0.5 m (location C for \"car\") with the SPL spectrum displayed in Figure 1, or from a vehicle with a light-bar mounted siren at a height (h\({}_{\text{tx2}}\)) of 2.5 m (location A for \"ambulance\") with the SPL spectrum displayed in Figure 2.
* Barrier located at a distance (d\({}_{1}\)) of 3 m from the sound source and a height (h\({}_{\text{bar}}\)) of 3 m with a maximum width (w) of 0.4 m (including their single or double cap on the top). * Potential sound receivers (R\({}_{\text{x1}}\) and R\({}_{\text{x2}}\)) positioned at a distance (d\({}_{2}\)) of 7 m from the barrier at heights of h\({}_{\text{Rx1}}\) = 1.7 m (location P for \"person\") and h\({}_{\text{Rx2}}\) = 5 m (location B for \"balcony\"), respectively. Therefore, the four schemes are as follows:  
* Configuration 1 (C-P), 'Car-Person'. * Configuration 2 (C-B), 'Car-Balcony'. * Configuration 3 (A-P), 'Ambulance-Person'. * Configuration 4 (A-B), 'Ambulance-Balcony'. Figure 4: Siren vehicle configurations (’3’, person and ‘4’, balcony). Figure 3: Car configurations (’1’, person, and ‘2’, balcony). It should be noted that to properly analyze the performance of the acoustic barriers, they must be tall enough to block the line-of-sight (LoS) from the road to the receiver. This can be achieved with a barrier of a relatively low height when the ground profile is not steeply sloped, the receiver is a person on the ground, and/or the acoustic barrier is close to the hard shoulder of the road (where the noise source is located). It should be explained here that the LoS arises between the source and the listener in the Ambulance-Balcony configuration and should thus be considered when testing the effect of the attached caps in this scenario. In the U.S., the average noise barrier heights vary widely from 2.1 m to up to approximately 5.5 m [41]. Across Europe, the minimum permissible height ranges from 1.0 m to 2.5 m, but the maximum permissible heights can vary from 4 m to 10 m or even higher, making the overall variations in barrier height seen in Europe far more significant than that seen in the U.S. [31]. Given the above, a moderate barrier height of 3 m was applied in the simulations of the present work. There are several reasons for this choice: it is cost-effective in terms of the investment required to build barriers of this height; it results in an aesthetic benefit for the surroundings; this height falls well within the average permitted noise barrier heights in both Europe and the U.S.; and it is a common height applied in studies undertaken in Japan (e.g., [PERSON] [42]). For each of the four propagation configurations analyzed, five reflective barrier cap models were considered. The cross-sections of each cap model are illustrated in Figure 5.  
Additionally, further analysis of the configurations illustrated in Figures 3 and 4 was conducted with barrier arrangements formed by two caps combined and resized to adhere to the maximum structure envelope size (total width of w and a height of w/2) and leaving a free space between them of w/4, as shown in Figure 6.  
There are 25 possible structural combinations of double-cap barriers (5\({}^{2}\)). It should be noted that the internal reflections between barrier caps over the mid support structure were intentionally not considered as their influence on the final pressure level is negligible. Therefore, a total of 100 simulations for the 25 double-cap barrier combinations and the four configurations of sources and receivers, plus 20 simulations for all single-noise barrier scenarios, besides four simulations for single screen barriers (as a base reference), were conducted, meaning  
Figure 5: Barrier caps considered: (1) Y-shaped; (2) rectangular; (3) T-shaped; (4) cylinder with radius w/2; and (5) trapezoid-shaped.