## 3 System Model
### Energy Consumption Model  
Due to the different media between the underwater environment and the air, the traditional energy consumption in TWSN could not be applied to UWSN. To ensure the reliability of the communication, it may be necessary to send data multiple times, which accounts for a larger proportion of the entire energy consumption because of the additional propagation loss. Thus, the energy expended to transmit the \(l\)-bits message (\(E_{tx}\)) and to receive this message (\(E_{rx}\)) are, respectively, [21]:  
\[E_{tx}=l\cdot E_{elec}+\frac{l}{R}\cdot P_{t} \tag{1}\]  
\[E_{rx}=l\cdot E_{elec} \tag{2}\]  
where \(E_{elec}\) is the electronics energy dissipated per bit, \(R\) denotes the transmission rate (bit/s), and \(P_{t}\) is the transmitted power. Specially, \(l/R\) expresses the time for sending the message. Generally, there are two main acoustic signal propagation mechanisms (namely, Urick Propagation Model [22]) as the geometrical effect. One is cylindrical spreading, and the other is spherical spreading. Cylindrical spreading refers to sound propagation in the shallow water (i.e., depth lower than 100 m) with a cylinder bounded by the surface and the water bottom, while spherical spreading is for sound propagation between the sender and the receiver in the deep water (i.e., deeper ocean). According to the assumed network model, we focus on the cylindrical spreading for shallow water in this paper. Let \(I_{t}\) be the current intensity, \(A\) be the cylindrical flank area, and then the transmitted power can be:  
\[P_{t}=A\cdot I_{t}=2\pi r\cdot H\cdot I_{t} \tag{3}\]  
where \(r\) and \(H\) are the radius and the height of the cylinder, which refers to the distance between acoustic source and receiver as well as water depth, respectively. To obtain \(I_{t}\), the average intensity \(I\), which is a plane wave with the root-mean-squared pressure \(p\) in a medium of density \(\rho\) and sound speed \(c\), should be known previously, where \(I=p^{2}/\rho c\)[23], and \(\rho c\) is the acoustic impedance. Let \(SL\) denote the source level, the original \(I_{t}\) can be expressed as the product of the intensity of the source level and the average intensity:  
\[I_{t}=10^{SL/10}\cdot\frac{p^{2}}{\rho c} \tag{4}\]  
Specially, \(\rho c\) is \(1.5\times 10^{6}\) kg/(m\({}^{2}\)s) in some underwater environments. Thus, a plane wave of root-mean-squared pressure (\(10^{-6}\) Pa) has an intensity of \(0.67\times 10^{-18}\) W/m\({}^{2}\)[23; 24], and from Equation (4), we can get \(SL\) within logarithmic mode:  
\[SL=10\log\left(\frac{I_{t}}{0.67\times 10^{-18}}\right) \tag{5}\]On the other hand, the source level can also be written by the passive sonar equation [21]:  
\[SL=SNR+TL+NL-DI \tag{6}\]  
where \(SNR\) refers to the signal to noise ratio, and \(TL\) and \(NL\) are the transmission loss under the underwater environment and the noise level (i.e., ambient noise caused by turbulence, shipping, waves and thermal noise), respectively, \(DI\) denotes the directivity index and is 0 while using the omni-directional hydrophones. Basically, the transmission loss (\(TL\)), which can be defined as the accumulated decreasing in acoustic intensity when an acoustic wave propagates outwards from the source [21], is a significantly important effect on sound communication in underwater. It can always be estimated by a variety of phenomena in underwater, e.g., geometrical spreading, absorption and scattering. With cylindrical spreading in shallow water, \(TL\) can be approximated as follows [24]:  
\[TL=10\log(r)+\alpha(f)\cdot r\times 10^{-3} \tag{7}\]  
where \(\alpha(f)\) refers to the absorption loss in medium using [PERSON]'s equation [23].