## 3 System Model and Description
### System Models of Existing and Proposed Protocols
#### 3.3.1 Forwarder Model  
Propagation idea of the proposed protocols and acoustic signals with their absorption constraints in UWSN environment are discussed below.  
Channel Fading  
The sudden change in signal strength over a distance '\(D\)' and frequency '\(F\)' is calculated using Equation (1) [21].  
\[A(D,F)=D^{k}\times\alpha(F^{D}), \tag{1}\]  
here, '\(\alpha(F)\)' is the absorption coefficient, while signal range define the signal geometric propagation which may be cylindrical, practical or spherical. In [13], the authors define '\(\alpha\)' using Equation (2).  
\[\alpha(F^{D})=\frac{0.11F^{2}}{1+F^{2}}+\frac{44F^{2}}{4100+F^{2}}+2.75\times 1 0^{-4}F^{2}+0.003, \tag{2}\]  
Figure 4: System model of BFS-SPF-Three.  
Figure 5: Legends of above Discussed Protocols.  
here, 'F' is measured in KHz and 'a(F)' is measured in dB/Km, respectively. In sea water, different noise resources (shipping noise, thermal noise and turbulence noise) exits and denoted by 'NR'. For the given frequency 'F', the overall noise power denoted by 'NP' with spectral density is calculated using Equation (3) [22]. Figure 6 shows the relationship of 'a\((F)\)' and 'F' using [PERSON]'s model. According to [PERSON]'s model, absorption of acoustic signals grows with the growth of 'F'.  
\[NP(F)=\sum_{i=1}^{NR}NP_{i}(F), \tag{3}\]  
'\(NP_{i(F)}\)' is measured in dB and it should be in between (1-NR). This attenuation relates directly with the distance of source from destination. This attenuation over 'D' is calculated using Equation (4).  
\[NP_{i}^{D}(F)=\sum_{i=1}^{NR}\frac{NP_{i}(F)}{D^{k}\alpha(F^{D})}. \tag{4}\]  
Channel Capacity is denoted by 'C' and it is the number of bits without error using bandwidth 'B'. The signal to noise ratio 'SNR' and the actual capacity is calculated using Equations (5) and (6). Whereas, packet error rate is denoted by 'PER'.  
\[C=B.\log_{2}(1+SNR), \tag{5}\]  
\[Actual\ Capacity(PER)=\frac{C}{1-H_{2}(PER)}, \tag{6}\]  
here, '\(H_{2}\)'is a binary entropy function using PER, which is defined as \(H_{2}=(PER-1)\log_{2}(1-PER)(PER\log_{2}PER)\).  
'\(E_{t}\)' represents the transmission energy and '\(E_{r}\)' is used to denote receiving energy. Equation (7) is used to calculate '\(E_{t}\)' and Equation (8) is calculating '\(E_{r}\)'. Whereas, transmission power is denoted as '\(P_{t}\)', reception power as '\(P_{t}\)', distance from \(n-1\) to \(n\)th node as \(D\) and data rate as '\(DR\)', respectively. Where, '\(N\)' is the total number of data packets '\(DP\)' and '\(THCs\)' are represents the total hop counts.  
Figure 6: Absorption and Frequency Relation.  
In addition, packet length is denoted as '\(PL\)',, total leftover energy of the network nodes as '\(E_{t}^{remaining}\), and total receiving energy of the network nodes after receiving the data packets as '\(E_{r}^{remaining}\). The '\(E_{t}^{remaining}\)' and '\(E_{r}^{remaining}\), are calculated using Equations (9) and (10).  
\[E_{t}=\frac{P_{t}\times PL}{DR}\times D, \tag{7}\]  
\[E_{r}=\frac{P_{r}\times PL}{DR}\times D, \tag{8}\]  
\[E_{t}^{remaining}=\sum_{DP=1}^{N}\sum_{HC=1}^{THC}n_{i}-E_{t}, \tag{9}\]  
\[E_{r}^{remaining}=\sum_{DP=1}^{N}\sum_{HC=1}^{THC}n_{i}-E_{r}. \tag{10}\]  
Initial energy is represented as '\(E_{Initial}\)' and total energy of the network as '\(E_{Total}\)'. Total energy is calculated using Equation (11).  
\[E_{Total}=E_{Initial}-(E_{t}^{remaining}+E_{r}^{remaining}). \tag{11}\]