## 3 System Model
### Directional Model in UASNs
#### 3.1.2 Optimal Beamwidth  
As a rule of thumb, the more communication nodes coveraged by a sponsoring node, the better; the fewer nodes out of its coverage, the better. This ensures directional network connectivity as much as possible. Assuming that some nodes are uniformly distributed in the Euclidean plane and that their coverage area is divided into 8 sectors, the number of nodes \(nCov(p)\) that an arbitrary node \(p\) covers can be denoted by the area that \(p\) covers, as the formula shows below:  
\[nCov(p)=S(-\frac{\pi}{8}<p<\frac{\pi}{8})=\int_{-\frac{\pi}{8}}^{\frac{\pi}{8} }dist(\theta)d\theta \tag{4}\]  
\[nExc(p) =S(\frac{\pi}{8}<p<\pi)+S(-\pi<p<-\frac{\pi}{8}) \tag{5}\] \[=\int_{\frac{\pi}{8}}^{\pi}dist(\theta)d\theta+\int_{\pi}^{-\frac {\pi}{8}}dist(\theta)d\theta\]  
where \(dist(\theta)\) denotes the farthest distance a beam can reach in direction \(\theta\), and Equation (5) denotes the uncovered area of \(p\); thus, the node coverage ratio(NCR) can be defined as a  
Figure 1: Normalized directional characteristics graphs for different combined coefficients.  
ratio of the coverage area of \(p\) to the uncovered area of \(p\), which is denoted by Equation (6) as follows:  
\[NCR(p) =\frac{nCov(p)}{nExc(p)} \tag{6}\] \[=\frac{S(-\frac{\pi}{8}<p<\frac{\pi}{8})}{S(\frac{\pi}{8}<p<\pi)+S(- \pi<p<-\frac{\pi}{8})}\] \[=\frac{\int_{-\frac{\pi}{8}}^{\frac{\pi}{8}}dist(\theta)d\theta}{ \int_{\frac{\pi}{8}}^{\pi}dist(\theta)d\theta+\int_{\pi}^{-\frac{\pi}{8}}dist( \theta)d\theta}\]  
Then the goal is to find the parameters that maximize \(NCR(p)\). Observing the fact that Equation (6) is closely related to \(dist(\theta)\), according to sonar equation [27], we achieve the following:  
\[\left\{\begin{array}{l}SL(0)-TL(dist(0))-NL=DT\\ SL(\theta)-TL(dist(\theta))-NL=DT\end{array}\right. \tag{7}\]  
In Equation (7), \(SL(\theta)\) denotes the sound source level in direction \(\theta\), \(TL(dist(\theta))\) denotes propagation loss, and \(NL\) and \(DT\) denote the noise level and directional index, individually. Therefore, a further simplification can be made as shown in Equation (8):  
\[SL(0)-TL(dist(0))=SL(\theta)-TL(dist(\theta)) \tag{8}\]  
Consequently, according to the relationship between the sound source level and sound pressure, and the empirical formula of propagation loss in shallow water, we can obtain the following:  
\[SL(\theta)=20 lg\frac{p(\theta)}{p_{ref}} \tag{9}\]  
\[TL(dist(\theta))=15\lg dist(\theta) \tag{10}\]  
Substituting Equations (9) and (10) into Equation (8), we obtain the following:  
\[dist(\theta)=dist(0)(\frac{p(\theta)}{p(0)})^{\frac{4}{3}}=dist(0)(R(\theta))^{ \frac{4}{3}} \tag{11}\]  
So, it is equivalent to solving the following optimization problem, combining the goal as follows:  
\[\left\{\begin{array}{l}\max NCR=\int\limits_{-\frac{\pi}{8}}^{\frac{\pi}{8}} \left(R(\theta)\right)^{\frac{4}{3}}d\theta\Bigg{/}(\frac{\pi}{8}\left(R( \theta)\right)^{\frac{4}{3}}d\theta+\int\limits_{-\pi}^{-\frac{\pi}{8}}\left( R(\theta)\right)^{\frac{4}{3}}d\theta)\\ s.t.\quad R(\theta)=(1+A_{1}\cos\theta+A_{2}\cos 2\theta)/(1+A_{1}+A_{2})\\ 0\leq A_{1}\leq 2\\ 0\leq A_{2}\leq 2\end{array}\right. \tag{12}\]  
Finally, the result of \(A_{1}=1.8305\), \(A_{2}=1.3019\) is achieved. The result maximizes \(NCR(p)\) and its directional performance is shown in Figure 2.  
The corresponding \(3 dB\) beamwidth is about \(2\theta_{-3 dB}=1.23 rad=70^{\circ}\), which is just slightly higher than the critical point of \(\alpha=\pi/3\) that guarantees connectivity, so the fan angle of \(\alpha=\pi/3\) is chosen as the breakpoint due to the number of beams is an integer and the beams do not overlap each other.