## 3 Results and Discussion
### Impact of Acoustic Transmission Characteristics on Acoustic Signal Propagation
#### 3.4.1 Acoustic Ray Bending  
We first studied the direct acoustic signal from the base station to the user, excluding the reflected signal from the seafloor or the sea surface. Due to acoustic ray bending, there was an acoustic shadow where the sound signal could not reach. By finding the critical reversal point of the bending ray, where the ray grazed the horizontal, we referred to the horizontal distance between the reversal point and the base station as the maximum service range. For the fixed base station, when the user was within the maximum range, the acoustic signal could reach the user directly by adjusting the initial grazing angle; when the user was outside the maximum range, there existed the acoustic shadow. Therefore, we could investigate the impact of acoustic ray bending on the maximum service range. The relationship between the initial grazing angle \(\alpha_{source}\) of the acoustic signal at the base station and the grazing angle \(\alpha_{reversal}\) at the reversal point is as follows [40]:  
\[ \begin{array}{l}p=\frac{\cos\alpha_{source}}{c_{source}}=\frac{\cos\alpha_{ external}}{c_{reversal}}\\ \alpha_{source}=\arccos\frac{c_{source}}{c_{reversal}}\quad\quad(\alpha_{reversal }=0)\end{array}  \tag{9}\]  
where \(p\) is the ray constant, \(c\) is the sound speed, and \(\alpha\) is the angle of the ray with respect to the horizontal, for which the upward angle is negative and the downward angle is positive. Using the initial grazing angle \(\alpha_{source}\), the horizontal distance \(x\) between the reversal point and the base station is calculated by the 2-D planar ray-trace method, as follows [40]:  
\[x=\sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}\frac{\left[1-\left(pc_{i}\right)^{2} \right]^{1/2}-\left[1-\left(pc_{i+1}\right)^{2}\right]^{1/2}}{pg_{i}} \tag{10}\]  
where \(g\) is the gradient of the sound speed, \(n\) is the number of layers from the source to the reversal point. Figure 7a shows the acoustic propagation path of the base station at different depths, and acoustic rays graze the sea surface and the seafloor, respectively. For the rays that grazed the sea surface, fixing the base station and decreasing the initial grazing angle of the upward acoustic signal, the ray could reach the area within the maximum service range on the sea surface. In contrast, when the initial grazing angle of the upward acoustic signal increased, the reversal point was away from the sea surface, and the ray could not reach the area between the reversal point and the sea surface. For the ray that grazed the seafloor, the situation was similar to that of the ray that grazed the sea surface. When the base station was fixed, the ray could reach the area within the maximum service range on the seafloor by increasing the initial grazing angle of the downward acoustic signal. In contrast, when the initial grazing angle of the upward acoustic signal decreased, the reversal point was away from the seafloor, and the ray could not reach the area between the reversal point and the seafloor. Figure 6: Simulated ocean environment with a calm sea surface, flat seafloor, and Munk sound speed profile. Figure 7b shows the maximum service range on the sea surface and the seafloor, and the maximum range varied with the depth of the base station. For the maximum service range on the sea surface, it gradually increased with increasing depth. Decreasing the height of the subsurface buoy could increase the maximum service range on the sea surface, and the maximum service range of the seafloor base station was close to 15 km. For the maximum service range on the seafloor, the ray grazed on the seafloor when the sound speed satisfied \(c_{source}\leq c_{reversal}\) according to Equation (9). For the Munk sound velocity profile, the sound speed in the water shallower than 400 m could not satisfy the condition, and the rays from the depth of 0-400 m were reflected or absorbed at the seafloor. When the depth was greater than 400 m, the sound speed satisfied the condition, and the ray could graze on the seafloor.