## 3 Proposed Protocol Design Scheme
### Energy Consumption and Network Model
The Euclidean distance (\(u\), \(V\)) between two sensor nodes is bound when the distance between them is constrained by \(u(h)\ \leq\ \delta(u,\ V)\ \leq\ V(h)\). Two sensor nodes have the same minimum hop distance (\(h\)). Because of this, network density \(\rho\) has a significant impact on boundary quality. In specific for each case where \(h>0\) obtains  
\[lim\ V(h)-u(h)=r_{min} \tag{2}\]  
where \(r_{min}\) shows the lowest communication range of the sensor nodes, a wide range of physical and theoretical characteristics can be seen in sensor networks. Therefore, a wide range of models is created based on the application's needs and the device's capabilities [20]. However, in most sensor nodes, sensing ability is reduced when the distance between the sensors and the objects increases. The general model for sensing in underwater wireless sensor networks is shown as:  
\[d\ (s,\ p)=\frac{\lambda}{[d\ (s,\ p)]k} \tag{3}\]  
where \(d\ (s,\ p)\) the Euclidean distance between \(s\) and the point \(p\), \(k\) is the sensor parameter, and \(\lambda\) is the positive constant in [20]. We consider that all the sensor nodes have low battery capabilities and are not recharged or replaced after implementation. The network's lifetime is described as the time it takes for the first sensor node in the network to run out of energy. The simulation area is divided into four equal squares labeled upper right square (URS), upper left square (ULS), bottom right (B.R. ), and bottom left (B.L.). Sink nodes (S.N.) move in a three-cornered path to gather data from the sensor nodes in each portion. Randomly installed nodes sense the characteristics and transform them into packets. So that they can be used in consequent processing steps, the data are sent to the sink node. Each of the nodes has direct communication with the sink nodes. Transmission mechanisms of acoustic signals differ between shallow and deep water. There are two ways to describe the audio signal transmission in shallow-water and deep-water spherical diffusion, and the energy consumption is caused by spherical distribution with water absorption. The signal-to-noise ratio of an acoustic signal on the receiver side can be calculated as [21]  
\[SNR=SL-TL-NL+DI \tag{4}\]  
\(TL\) is the transmission loss (dB), \(NL\) is the noise level, \(SL\) is the receiver source, and \(DI\) is the directivity index. Calculate the transmission loss of circular spread signals, then it is calculated by  
\[TL=10 log_{2}\delta(u,\ V)+\alpha\delta(u,\ V)\times 10^{-3} \tag{5}\]  
Here, (\(u\), \(V\)) is the distance between the sender and receiver and is the medium-dependent frequency coefficient in dB. [PERSON]'s empirical formula gives the absorption loss in [22]. \[\alpha=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^{2}}+2.75\times 10^{-4}f^{ 2}+0.003 \tag{6}\]  
Here, \(\alpha\) is in dB/Km and \(f\) is in KHz. Waves, shipping, wind, and considerable mammal activity can alter NL's noise level in shallow water. For simplicity, an average noise level NL in shallow water is taken as 70 decibels [22]. The following statement can be used to express the relationship between the transmitted signal intensity at one meter from the source and the radiated sound intensity in decibels [23]:  
\[SL=10\ log_{2}\ \frac{I_{t}}{1\ \mu\ Pa} \tag{7}\]  
Here, \(I_{t}\) is in \(\mu\ Pa\) and solving for \(I_{t}\) Yields. \[I_{t}=10^{5L/10}\times 0.67\times 10^{-18} \tag{8}\]  
Thus, the \(P_{t}\) achieves intensity is the transmitter power. Distance from transmitter to receiver is calculated as 1 m at this point [23]. \[P_{t}=2\pi\times H\times I_{t} \tag{9}\]  
Here, \(H\) is the depth of water in meters, and \(P_{t}\) is in watts. The function parameters (energy, distance, and bit error rate) are considered to select a destination. The packet arrives directly at the sink node from a neighboring source node. Otherwise, the packet is sent by the source node considering the multihopping. The parameters for selecting a destination where data will most likely be forwarded are listed below.