## 1 Introduction
Specifically, as the angle of transmission changes from 0 to \(10^{\circ}\), we observe that horizontal range to a LEP grows, indicating that for a transmission angle more than \(4^{\circ}\), only few iterations are needed to determine the number of LEPs within the transmission range of a node. Moreover, for transmission angles less than \(4^{\circ}\), a node may resort to a straight-line model as the variation in angle is minimal. Beam Selection Challenge: To establish an underwater communication link, node \(n_{T}\) typically makes an omni-directional transmission to reach a certain receiver node \(n_{R}\). Node \(n_{R}\) often detects multiple copies of the transmitted acoustic signal by \(n_{T}\) due to the multipath effect where the path of each received signal is further aggravated by refraction experienced due to SS variations along its propagation path. Such multipath effect suggests that when directional transmission is used, a communication link to a neighbor can be established using multiple angles. Moreover, in shallow water scenarios, refraction along the path is often ignored and the multipath signals are classified into four types, namely direct, surface reflected, bottom reflected and surface bottom reflected rays [6]. On the other hand, the deep-water multipath signals are quite different where losses experienced by the surface bottom reflected rays are often high; making these signals nondetectable due to the large separation between both surfaces. Moreover, when the acoustic signal moves from a lower SS region to a higher one, refraction may cause a propagation directional reversal along the depth generating a LEP along the propagation path, as illustrated in Figure 1a. As shown in Figure 1b, different ray types can be established among communicating pairs in deep-water setups where a transmitted beam may cover more than one type. Moreover, selecting two overlapping beams may result in utilizing the same ray type in establishing the communication link and thus wasting the energy of the transmitter. Furthermore, selecting disjoint beams may result in missing the ray types connecting the pairs and thus placing the neighbor in the shadow zone. Given these challenges and the fact that nodes in AUNsare usually sparsely located [3], a method is required to govern the selection of transmitted beams to ensure minimal overlaps while avoiding shadow zones, as discussed next. Shadow zones occur in two different conditions. The first condition is encountered when nodes select transmission angles that form disjoint beams, i.e., the angle between neighboring beam edges is greater than zero. An example of such shadow zone is shown in Figure 2, where the Bellhop ray tracing algorithm is utilized to determine the propagation path of selected beams [7]. In Figure 2, when using beams 3 and 4, nodes located in the dotted region will not hear the message sent by the transmitter. Such beam selection results in generating a shadow zone in the gaps observed between beams. The second shadow zone condition is related to the underwater losses where the losses are fundamentally due to the absorption and spreading of acoustic wave where spreading dominates in long-ranges [8]. Since each transmitted acoustic beam experiences unique refraction conditions, the distance at which the shadow zone is created is dependent on the spreading pattern. To illustrate this fact, consider beam 4 in Figure 2 where the loss rate rapidly increases due to the divergence of beam edges. Thus, when the spreading reaches a point where the signal power drops below the receiver sensitivity, a shadow zone is created. Since nodes residing in the shadow zones cannot detect transmissions sent by neighboring nodes, avoiding shadow zones for nodes becomes inevitable to form a connected network. On the other hand, the overlapping beams can be categorized into two types. The first, shown as the striped region (Zone 1) in Figure 2, is realized when selecting adjacent beams with their center angles deviation less than their beam widths.