# Adaptive Power-Controlled Depth-Based Routing Protocol for Underwater Wireless Sensor Networks
## 4 Theoretical Analysis and Experimental Results
### Theoretical Analysis  
The main purpose of the proposed protocol is to reduce energy consumption through a power control mechanism and balance energy consumption by considering residual energy information. The rationale behind the power control mechanism's ability to reduce energy consumption can be primarily attributed to two aspects. Firstly, in an underwater acoustic network, the distance between each pair of connected nodes is not always equal to the maximum transmission distance of the acoustic signal. Consequently, there exists some redundancy in the transmission power of the nodes, which can be reduced through the implementation of the power control mechanism. This point is relatively straightforward to comprehend. Secondly, the power control mechanism enables the decomposition of long-distance single-hop transmissions into multiple short-distance hops. It is well known from reference [17] that multi-hop transmission can effectively reduce energy consumption. However, the article does not provide formal proof for this claim. In [18], the authors conducted an analysis of the scenario with only one relay node and demonstrated that under certain conditions, multi-hop transmission is more energy efficient than single-hop transmission. However, in practical scenarios, there is typically more than one relay node involved in data forwarding. Therefore, in the subsequent investigation, we will consider  
Figure 6: Localized recovery example. the more general scenario where multiple nodes participate in relaying. We will analyze the effectiveness of the power control mechanism in reducing energy consumption, thereby demonstrating the effectiveness of our designed routing protocol. Case 1: When the relay node is deployed on the straight line between the source and the destination. When there is only one relay node, we can find the energy consumption for this scenario in [18] as follows:  
\[E_{1\_node}=\frac{LP_{0}}{Ra}(d_{SR_{1}}^{k}a(f)^{d_{SR_{1}}}+d_{R_{1}D}^{k}a(f )^{d_{R_{1}D}}) \tag{13}\]  
Here, \(E_{1\_node}\) represents the total energy consumption in the case of one relay node. The energy consumption of direct transmission is:  
\[E_{direct}=\frac{LP_{0}d_{SD}^{k}a(f)^{d_{SD}}}{Ra} \tag{14}\]  
Among them, \(P_{0}\) is calculated by Equation (4). Since the receiving power and idle power are much smaller than the transmitting power, we neglect them here. The authors in [18] have already demonstrated that when the relay node is not located at the source or destination node, i.e., when it is positioned between the source and destination nodes, we have the following relationship:  
\[E_{1\_node}<E_{direct} \tag{15}\]  
As is shown in Figure 7, when there are \(n\) relay nodes, the energy consumption can be represented as:  
\[E_{n\_node}=\frac{LP_{0}}{Ra}(d_{SR_{1}}^{k}a(f)^{d_{SR_{1}}}+d_{R_{1}R_{2}}^{ k}a(f)^{d_{R_{1}R_{2}}}+\cdots+d_{R_{n-1}R_{n}}^{k}a(f)^{d_{R_{n-1}R_{n}}}+d_{R_{n }D}^{k}a(f)^{d_{R_{n}D}}) \tag{16}\]  
In this case, there are multiple variables in \(E_{n\_node}\), and a direct derivation would be quite complex. Therefore, we can use mathematical induction to prove that \(E_{n\_node}<E_{direct}\). We have already established that when \(n=1\), we have \(E_{direct}>E_{1\_node}\), i.e., \(E_{direct}>E_{SR_{1}}+E_{R_{1}D}\), \(E_{SR_{1}}\) and \(E_{R_{1}D}\) represent the communication energy consumption between node \(S\) and \(R_{1}\), and between \(R_{1}\) and \(D\), respectively;  
Figure 7: Example of Multi-hop Transmission with \(n\) Relay Nodes.